5?RKEIEY 

1    LIBRARY 

UNIVERSITY  OF 
CALIFORNIA 

EARTH 


SCIENCES 
LIBRARY 


MANUAL  OF 
PETROGRAPHIC  METHODS 


McGraw-Hill  BookCompany 


Pu/ifis/iers  ofB 

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MANUAL  OF 
PETROGRAPHIC  METHODS 


BY 


ALBERT  JOHANNSEN,  PH.  D. 

ASSISTANT   PROFESSOR   OF   PETROLOGY,    THE   UNIVERSITY   OF   CHICAGO 


McGRAW-HILL  BOOK  COMPANY,  INC. 
239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 
1914 


COPYBIGHT,  1914,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


THE. MAPLE. PRESS. YORK. PA 


SCIENCES 
i'BRARX 

PREFACE 

The  desire  of  an  increasing  number  of  students  for  more  complete  informa- 
tion in  regard  to  modern  petrographic-microscopic  methods  than  is  to  be 
found  in  any  English  work  on  the  subject,  has  led  the  author  to  prepare  this 
book.  While  the  preliminary  portions  of  many  excellent  elementary  and 
intermediate  text-books  on  optical  mineralogy,  and  certain  portions  of  most 
crystallographies  and  mineralogies,  are  devoted  to  microscopic  methods, 
none  makes  any  pretense  at  completeness,  and  even  in  the  present  work  of 
over  600  pages  those  parts  devoted  to  microchemical  methods,  examination 
of  opaque  minerals  and  mineral  grains,  etc.,  might  be  expanded,  and  there 
might  be  added  chapters  on  photomicroscopy,  projection  apparatus  for 
polarized  light,  etc.,  etc. 

Owing  to  the  fact  that  many  students  who  take  up  the  subject  of  petro- 
graphy are  weak  in  their  preliminary  training  in  physics  and  mathematics, 
the  author  has  thought  it  best  to  treat  the  subjects  of  harmonic  motion,  light, 
and  lenses  somewhat  more  fully  that  he  otherwise  would  have  done.  The 
mathematical  demonstrations  may  have  a  somewhat  formidable  look  to  the 
non-mathematician,  but  this  is  due  more  to  the  fact  that  they  are  carried  in 
detail  through  the  various  steps,  and  consequently  are  more  easily  followed, 
than  that  they  are  actually  difficult.  Likewise  for  the  non-mathematician, 
the  more  cumbersome  algebraic  methods  have  been  used  in  certain  demon- 
strations rather  than  those  of  the  calculus. 

This  book  is  not  intended  for  beginners  working  without  instructors,  for  to 
such  the  great  variety  of  methods  described  will  only  bring  confusion.  The  in- 
vestigator and  advanced  student,  however,  should  be  familiar  with  all  methods, 
old  and  new;  for  methods,  once  abandoned,  may  serve  as  preliminary  stages 
to  new  lines  of  thought  and  further  improvements.  Unfamiliarity  with  what 
had  been  done  in  the  past  has  frequently  led  to  duplication  of  work,  perhaps 
with  a  considerable  expenditure  of  time  that  might  have  been  used  to  better 
advantage. 

The  data  for  this  book  have  been  brought  together  from  widely  scattered 
sources,  as  may  be  seen  from  the  footnotes.  Much  of  the  original  material 
is  in  foreign  publications,  inaccessible  to  the  majority  of  students.  It  has 
not  been  thought  sufficient  to  take  these  references,  even  for  the  general 
bibliographies,  at  second  hand,  but  the  original  works  have  been  consulted. 
In  every  case  where  the  reference  seemed  to  be  of  sufficient  importance  to 
insert  but  the  original  work  was  not  accessible,  the  footnote  has  been  marked 
with  an  asterisk  (*). 


VI  PREFACE 

In  this,  the  first  attempt  to  give  in  English  a  comprehensive  review  of 
petrographic  methods,  it  cannot  be  otherwise  than  that  there  should  be  many 
omissions — -it  is  to  be  hoped  not  many  errors.  If  any  is  found  the  author 
will  be  extremely  glad  to  have  his  attention  called  to  it,  as  he  will  also  be  for 
criticisms,  or  suggestions  for  additional  material  that  should  be  included. 

For  more  or  less  general  information  the  author  is  indebted  to  the  standard 
works  of  Rosenbusch  and  Wiilfing,  Duparc  and  Pearce,  Groth,  Iddings, 
Miers,  Tutton,  and  Wright.  For  permission  to  use  certain  figures  he  wishes 
to  express  his  thanks  to  Professors  Becke,  Duparc,  and  Miers,  and  Doctor 
Wright,  and  to  the  manufacturers  of  certain  apparatus  for  the  use  of  elec- 
trotypes. The  half-tones  of  the  interference  figures  are  reproduced  from  the 
late  Doctor  Hauswaldt's  magnificent  Interferencerscheinungen  im  polarisirten 
Lichte,  and  are  given  with  the  kind  consent  of  Frau  Hauswaldt.  The  author 
has  to  express  his  appreciation  to  the  attendants  at  the  John  Crerar  Library 
for  their  uniform  courtesy  in  obtaining  for  him  the  innumerable  volumes 
consulted  in  this  work,  and  especially  for  their  efforts  to  find  the  proper  vol- 
umes of  the  numerous  publications  to  which  wrong  citations  were  given  by 
other  writers.  To  Professor  G.  W.  Myers  he  is  indebted  for  certain  mathe- 
matical demonstrations.  Most  especially  he  desires  to  thank  Professor  A.  C. 
Lunn,  of  the  University  of  Chicago,  who  has  placed  him  under  great  obliga- 
tions by  critically  reading  those  parts  of  the  manuscript  dealing  with  lenses 
and  light,  and  for  giving  him  valuable  suggestions.  Finally  his  thanks,  and 
the  thanks  of  all  petrographers  to  whom  this  book  may  prove  useful,  are  due 
to  the  publishers  for  their  willingness  to  issue  a  work  of  this  kind,  which  must 
necessarily  have  a  limited  circulation. 

ALBERT  JOHANNSEN. 
THE  UNIVERSITY  or  CHICAGO, 
December  24,  1913. 


CONTENTS 

PAGE 

PREFACE v 

TABLE  OF  CONTENTS vii 

LIST  OF  ABBREVIATIONS xxiii 

CHAPTER  I 

MlXERALOGICAL    PRINCIPLES I 

Crystals i 

Crystallographic  axes i 

The  Weiss  parameters 3 

The  Naumann  system      3 

The  Miller  indices 3 

Zones .  4 

CHAPTER  II 

STEREOGRAPHIC  PROJECTION 5 

Introductory 5 

Definitions 5 

Locating  points 6 

Circles  drawn  upon  a  sphere  appear  as  circles  in  stereographic  projection   .    .  8 

Spherical  angles  appear  in  their  true  values  in  stereographic  "projection     .    .  9 

Graphical  solutions  of  problems 10 

Protractors  and  scales 14 

Calculating  the  location  of  points  in  stereographic  projection 19 

Accuracy  of  stereographic  projection .  22 

Problems  solved  by  means  of  a  stereographic  net  .    .    .    .    : 22 

Various  accessories  used  in  stereographic  projection 25 

CHAPTER  III 

A  FEW  PRINCIPLES  OF  OPTICS 29 

The  nature  of  light 29 

Corpuscular  or  emission  theory 29 

The  undulatory  or  wave  theory  of  Huygens 30 

The  electromagnetic  theory 30 

The  ether 31 

Wave  motion 31 

The  movements  of  oscillation 31 

Simple  harmonic  motion 33 

Isochronism  and  angular  velocity 33 

Harmonic  curves 34 

Combinations  of  simple  harmonic  motions       37 

Combinations  of  harmonic  curves 41 

vii 


viii  CONTENTS 

CHAPTER  IV 

PAGE 

ISOTROPIC  MEDIA 48 

Definitions 48 

Wave  motion  in  isotropic  media 48 

Intensity  of  light       49 

Color  of  light 49 

Velocity  and  wave  length  of  light 49 

Wave  front  and  wave  surface 49 

Reflection  of  waves 50 

Passage  of  light  into  a  medium  of  different  density 52 

Refraction  of  light  upon  passing  into  an  isotropic  medium  of  different  density  52 

Index  of  refraction 54 

Passage  of  light  into  different  isotropic  media 54 

Relation  between  indices  of  refraction  and  velocity  of  propagation  of  light  .  56 

Total  reflection  and  the  critical  angle 56 

Polarization,  and  light  polarized  by  reflection 57 

Angle  of  polarization 58 

Variation  in  intensity — Malus'  law 59 

Polarization  by  refraction 59 

Arago's  law 60 

CHAPTER  V 

ANISOTROPIC  MEDIA 61 

Single  refraction  and  double  refraction 61 

Optically  uniaxial  crystals 62 

Double  refraction  in  calcite 62 

Optic  axis 70 

Principal  optic  section 70 

Positive  and  negative  uniaxial  crystals 70 

Velocity  of  any  intermediate  ray 71 

Velocity  of  any  intermediate  wave 72 

Vibration  directions      73 

Ray  surface  and  wave  surface 75 

Graphic  development  of  ray  and  wave  surfaces 76 

Curve  of  ease  of  vibration  (Fresnel's  curve  of  elasticities) 79 

Fresnel's  ellipsoid 80 

The  optical  indicatrix 80 

Huygens'  construction  for  double  refraction  in  uniaxial  crystals 82 

Summary  of  the  optical  properties  of  uniaxial  crystals 89 

CHAPTER  VI 

ANISOTROPIC  MEDIA  (continued) 91 

Optically  biaxial  crystals 91 

Vibration  axes 91 

Fletcher's  indicatrix 92 

Ray  surface 94 

Wave  surface 98 

Optic  biradials  or  secondary  optic  axes 99 


CONTENTS  ix 

PAGE 

Optic  binormals  or  primary  optic  axes 100 

Interior  conical  refraction 100 

Exterior  conical  refraction       102 

Optic  axial  angle,  true  and  apparent 102 

Equations  expressing  the  value  of  the  true  axial  angle 103 

Relation  between  the  true  and  apparent  axial  angles 104 

Plane  of  the  optic  axes 105 

Bisectrices 105 

Positive  and  negative  biaxial  crystals 105 

Polarization  by  double  refraction       106 

Circular  and  elliptical  polarization 107 

Rotary  polarization 108 

Summary  of  optical  principles no 

CHAPTER  VII 

LENSES 114 

Definitions *. 114 

Axis,  vertices  and  thickness  of  a  lens 114 

Optical  center 114 

Principal  focal  point 115 

Conjugate  foci  of  convex  lenses      116 

Refraction  through  simple  lenses 116 

Focus  of  combined  lenses 118 

Gauss'  method 119 

Application  of  Gauss'  cardinal  points  to  the  determination  of  the  image 

formed  by  a  lens 121 

Equations  for  the  determination  of  the  cardinal  points  of  any  lens  system   .    .    .  121 

Lateral  magnification 121 

Convergence  of  a  lens 122 

Formation  of  images  by  lenses 122 

System  of  two  faces      1 23 

Aberration      129 

Angular  aperture       131 

Numerical  aperture 131 

Table  of  numerical  apertures  for  various  angular  apertures      132 

Apertometer       132 

Magnifying  power 133 

CHAPTER  VIII 

THE  MICROSCOPE 136 

Simple  microscope 136 

Hand  lens 136 

Compound  microscope 138 

Formation  of  the  image 138 

Optical  and  mechanical  tube  lengths 138 

Focal  length 140 

Magnifying  power 140 

Field  of  view 140 

The  petrographic  microscope 141 


x  CONTENTS 

PAGE 

Description '....,•• 141 

The  mechanical  parts  of  a  petrographic  microscope  .    .    .    .    .    . 142 

Foot 142 

Pillar  or  post 142 

Limb  or  arm 142 

Stages,  simple  and  mechanical 142 

Verniers 144 

Body  tube .    :    .    .    .    .  145 

Objective  holders 146 

Slot  for  accessories 148 

Centering  device  for  objective ...    ."   .    .    /-  ....  148 

Coarse  and  fine  adjustment 149 

Sub-stage 150 

Diaphragms 151 

CHAPTER  IX 

THE  MICROSCOPE  (CONTINUED) 154 

The  Optical  Parts  of  a  Petrographic  Microscope 154 

Illuminating  apparatus 154 

Polarizing  prisms 158 

Introduction 158 

Nicol  prism 158 

Sang  prism .    .    .    .  164 

Foucault  prism 165 

Hartnack-Prazmowski  prism 165 

Talbot  prism 167 

Glan  prism 167 

Thompson  prisms ...  167 

Fuessner  prisms 168 

Bertrand  prisms 169 

Ahrens  prism  (1884) 170 

Madan  prism 171 

Ahrens  prism  (1886) 171 

Grosse  double-slit  air  prism 172 

Leiss  prism 172 

Von  Lommel  prism 173 

Von  Fedorow  polarizer ...  173 

Halle  prisms 174 

Glass  polarizing  prisms 174 

Summary  of  properties  of  polarizing  prisms 175 

Polarizer  and  analyzer 176 

Determination  of  the  vibration  directions  in  the  nicol  prisms 178 

Bertrand  lens 178 

CHAPTER  X 

THE  MICROSCOPE  (CONTINUED) 180 

The  optical  parts  of  a  petrographic  microscope  (continued) 180 

The  objective 180 

Introductory 180 


CONTENTS  xi 

PAGE 

Definition 180 

'Depth  of  definition  (depth  of  focus)  or  penetration 180 

Flatness  of  field 181 

Illuminating  power 181 

Resolving  power 181 

Working  distance 182 

Magnifying  power 182 

Dry  and  immersion  objectives 183 

Classification  of  objectives  according  to  correction  for  aberration 185 

Effect  of  cover-glasses  of  different  thicknesses  upon  objectives 185 

Comparative  table  ol  dry  achromatic  objectives  of  different  makers  ....  189 

Aperture  table 190 

Testing  the  objective 191 

Cost  of  objectives 193 

The  ocular  or  eyepiece 193 

Huygens  ocular 193 

Ramsden  ocular 194 

Compensating  oculars 194 

Comparative  table  of  Huygens  oculars  of  different  makers 195 

Oculars  for  special  purposes 195 

Demonstration  oculars 196 

Focussing  cross- hairs  in  the  ocular  .  .  .  • 197 

Replacing  cross-hairs 197 

Magnification  of  the  compound  microscope 197 


CHAPTER  XI 

VARIOUS  MODERN  MICROSCOPES 199 

Introduction 199 

Leitz  stand  AM 199 

Leitz  Berkey  model 200 

Leitz  new  stand 200 

Seibert  microscope 202 

Fuess  stand  Via 202 

Fuess  stand  Ilia 203 

Fuess  microscope,  Model  Ib 205 

Fuess  microscope,  Model  Ha • 207 

Zeiss  crystallographic  and  petrographic  microscope  III  MD 208 

Zeiss  small  mineralogical  stand  VM 209 

Reichert  mineralogical  stand  MI 209 

Reichert  mineralogical  microscope  M  VIII 211 

Bausch  &  Lomb  LCH  petrographic  microscope 211 

Xachet  microscope 213 

Swift  improved  Dick  petrographic  microscope        215 

Swift  large  petrographic  microscope 215 

Beck  London  petrographic  microscope 217 

Societe  Genevoise  universal  microscope 218 

Fuess  microscope  for  the  theodolite  method 218 

Beck  Rosenhain  metallurgical  microscope 221 


xii  CONTENTS 

CHAPTER  XII 

PAGE 
SELECTING,  USING,  AND  TAKING  CARE  OP  A  MICROSCOPE ..222 

Selecting  a  microscope 222 

Use  and  care  of  a  microscope 223 

Light 223 

Table 225 

Method  of  working 225 

Position 225 

Proper  eye  to  use 225 

Eye  shade 226 

Amount  of  light 227 

Proper  magnifying  power  to  use 227 

Objective  clutch 227 

Focussing 227 

Changing  the  ocular 228 

Hints  on  the  care  of  a  microscope  . 228 

.  Care  of  stand 228 

Care  of  nicols  and  lenses 228 

Testing  and  adjusting  the  microscope  and  the  accessories 229 

Cross-hairs 229 

Bertrand  ocular 230 

Bertrand  lens,  Centering 230 

Nicol  prisms 230 

Accessories 23 1 

CHAPTER  XIII 

OBSERVATIONS  BY  ORDINARY  LIGHT 233 

Ordinary  light 233 

Determination  of  crystal  form 233 

Cleavage  and  parting 235 

Determination  of  refractive  indices 237 

Relief 237 

The  method  of  the  Due  de  Chaulnes 238 

Brewster's  method  for  determining  the  refractive  index  of  a  liquid 241 

Becquerel  and  Cahour's  method  for  determining  the  refractive  index  of  a 

liquid 241 

Bertin's  method 242 

Sorby's  method 244 

CHAPTER  XIV 

OBSERVATIONS  BY  ORDINARY  LIGHT  (CONTINUED) 249 

Determination  of  the  refractive  indices  of  a  mineral  by  the  immersion  or  embed- 
ding method 249 

Maschke 249 

Sorby 250 

Thoulet 250 

Stephenson 251 

Rohrbach 251 

Brauns 251 


CONTENTS  xill 

PAGE 

Bertrand 251 

Klein 252 

Schroeder  van  der  Kolk.. 252 

Zirkel 252 

Retgers 252 

Ambronn  (1893) 253 

Ambronn  (1896) 254 

Marpmann 254 

Schroeder  van  der  Kolk  (1898) 255 

Schroeder  van  der  Kolk  (1900) 256 

Immersion  fluids 259 

Determination  of  the  refractive  indices  of  fluids. 265 

Introductory 265 

Smith's  method 265 

Pauly's  method 266 

Michel-L6vy's  indicators 268 

De  Souza-Brandao's  indicators 268 

Clerici's  method 270 

CHAPTER  XV 

OBSERVATIONS  BY  ORDINARY  LIGHT  (CONTINUED) 271 

Determination  of  the  refractive  indices  of  a  mineral  by  the  Becke  method  ...  271 

Becke's  explanation 271 

Hotchkiss' explanation 272 

Grabham's  explanation 274 

Inclined  illumination 275 

Viola-deChaumes-Becke  method 276 

Viola-Becke  method 276 

Practical  applications  of  the  Becke  method 277 

Refractive  index  of  Canada  balsam 283 

Relation  between  refractive  index  and  density 285 

The  examination  of  opaque  minerals 285 

CHAPTER  XVI 

MEASUREMENTS  UNDER  THE  MICROSCOPE 287 

Measurement  of  enlargement 287 

Measurement  of  the  field  of  view 287 

Measurement  of  lengths 288 

Measurement  of  areas 290 

Measurement  of  thicknesses 293 

Measurement  of  plane  angles 293 

Measurement  of  optic  axial  angles : 295 

CHAPTER  XVII 

DRAWING  APPARATUS 296 

CHAPTER  XVIII 

ROTATION  APPARATUS 300 


xiv  CONTENTS 

CHAPTER  XIX 

PAGE 

THE  COLOR  OF  MINERALS 309 

Idiochromatic  and  allochromatic  minerals 309 

Determination  of  color 310 

Determination  of  the  color  of  opaque  minerals 311 

CHAPTER  XX 

MONOCHROMATIC  LIGHT 313 

The  production  of  monochromatic  light 313 

Ray  niters \    .........  314 

Incandescent  vapors  of  solids 316 

Incandescent  gases 317 

Dispersed  white  light  produced  by  a  monochromator 317 

CHAPTER  XXI 

EXAMINATION  BY  PLANE  POLARIZED  LIGHT 320 

Absorption,  dichroism,  pleochroism 320 

Absorption  of  light  in  crystals 320 

Isotropic  substances .....'...........  320 

Anisotropic  substances 320 

Uniaxial  crystals .;   .....  321 

Biaxial  crystals 322 

Pleochroic  halos 323 

Pseudo-pleochroism,  pseudo-dichroism,  or  pseudo-absorption 324 

Interference  phenomena,  without  the  analyzer,  produced  by  an  overlying 

pleochroic  mineral 324 

Determination  of  pleochroism 325 

Determination  of  the  absorption  coefficient 326 

CHAPTER  XXII 

INTERFERENCE  COLORS 328 

Interference 328 

Color  of  thin  plates 328 

Newton's  color  scale \    . 330 

Color  scale  according  to  Quincke 331 

Color  scale  according  to  Kraft 332 

CHAPTER  XXIII 

EXAMINATION  BETWEEN  CROSSED  NICOLS '. 336 

Isotropic  substances 236 

Anisotropic  substances 336 

Retardation  in  anisotropic  media 337 

Phasal  difference •    •    •  337 

Interference  of  polarized  light 337 

Extinction  angles 339 


CONTENTS  xv 

PAGE 
Passage  of  monochromatic  light  through  two  nicol  prisms  and  a  mineral 

section 34  * 

The  intensity  of  the  emerging  light 343 

Two  superposed  mineral  plates 346 

Examination  by  white  light.     Interference  colors 348 

Calculation  of  the  value  of  the  birefringence  in  any  section 351 

Lines  of  equal  birefringence 355 

Abnormal  birefringence 359 


CHAPTER  XXIV 

EXAMINATION  BETWEEN  CROSSED  NICOLS  (CONTINUED) 361 

Determination  of  the  vibration  directions  in  mineral  plates 361 

Optical  character  of  the  elongation 361 

Accessories  used  for  the  determination  of  the  vibration  directions  of  a  mineral  .  362 

Kinds  of  accessories 362 

Simple  plane  parallel  plates • 362 

Quarter  undulation  mica  plate 362 

Unit  retardation  plate 365 

Retardation  wedges 365 

Simple  quartz  of  gypsum  wedge 365 

Von  Fedorow  mica  comparator 366 

Wright  combination  wedge 366 

Johannsen  quartz-mica  wedge 367 


CHAPTER  XXV 

EXAMINATION  BETWEEN  CROSSED  NICOLS  (CONTINUED) 369 

Determination  of  the  order  of  birefringence 369 

Compensating  wedge  for  the  determination  of  birefringence 369 

Michel-Levy  chart  of  birefringences 370 

Babinet  compensator 373 

Von  Chrustschoff  twin  compensator 376 

Michel-LeVy  comparateur 377 

Von  Fedorow  method  for  determining  low  interference  colors      378 

Cesaro  wedge 379 

Amann  birefractometer 379 

Von  Fedorow  mica  comparator 379 

Salomon's  method  for  computing  the  value  of  &>-€  in  uniaxial  minerals    .    .    .  383 

Wallerant's  method  for  measuring  slight  double  refraction 383 

Nikitin's  method 383 

Joly's  method 383 

Wright  combination  wedge 383 

Evans  simple  quartz  wedge 383 

Evans  double  quartz  wedge 384 

Seidentopf  quartz  wedge  compensator 384 

Wright  double  combination  wedge 385 

Nikitin  quartz  compensator 385 


xvi  CONTENTS 

CHAPTER  XXVI 

PAGE 

EXAMINATION  BETWEEN  CROSSED  NICOLS  (CONTINUED) 386 

Determination  of  very  slight  double  refraction 386 

Sensitive  violet 386 

Biot  quartz  plate 386 

Savart  plate 386 

Soleil  bi-quartz  plate 387 

Bravais  twinned  mica  plate      387 

Klein  quartz  plate 387 

Bertrand  ocular 388 

Calderon  ocular 388 

Traube  bi-mica  plate 388 

Brace  half-shade  elliptical  polarizer  and  compensator 388 

Sommerfeldt  twinned  gypsum  plate 388 

Kb'nigsberger  ocular ;    .  388 

Half-shade  plates 388 

CHAPTER  XXVII 

EXAMINATION  BETWEEN  CROSSED  NICOLS  (CONTINUED) 390 

Practical  methods  for  the  determination  of  extinction  angles 390 

Relation  of  the  optical  ellipsoid  to  the  crystallographic  axes.     Parallel  and  in- 
clined extinction 390 

Methods  for  measuring 392 

Unit  retardation  plate 393 

Bravais  twinned  mica  plate 393 

Kobell  stauroscope 394 

Klein  quartz  plate 394 

Bertrand  ocular 394 

Calderon  plate 395 

Von  Fedorow's  method  by  means  of  the  universal  stage 395 

Wiedemann  double  double-quartz  wedge 395 

Stober  quartz  double  plate 396 

Traube  bi-mica  plate 396 

Mac6  de  Lepinay  half-shade  plate 396 

Sommerfeldt  twinned  gypsum  plate 397 

Wright  artificially  twinned  quartz  plate 397 

Wright  bi-quartz  wedge  plate 398 

CHAPTER  XXVIII 

EXAMINATION  BETWEEN  CROSSED  NICOLS  (CONTINUED) 399 

Calculation  of  extinction  angles  in  random  thin  sections 399 

Zones 399 

Calculation  of  extinction  angles  for  any  face  of  the  100-010  zone  of  a  mono- 
clinic  crystal 399 

Calculation  of  the  extinction  angle  for  any  face  of  any  zone  of  any  crystal,  403 

Graphical  methods  for  the  determination  of  extinction  angles  on  any  plane  .    .    .  406 

Extinction  diagram  and  curves  of  equal  extinction 410 

Influence  of  dispersion  upon  extinction  angles 412 


CONTENTS  xvii 

CHAPTER  XXIX 

PAGE 

OBSERVATIONS  BY  CONVERGENT  POLARIZED  LIGHT  ..." 413 

Polariscope,  conoscope 413 

Interference  figures 415 

Isotropic  crystals 415 

Random  sections 415 

Anisotropic  crystals 416 

Uniaxial  crystals 416 

Section  perpendicular  to  the  optic  axis 416 

Section  oblique  to  the  optic  axis 418 

Sections  parallel  to  the  optic  axis 419 

Biaxial  crystals      420 

Sections  cut  at  right  angles  to  the  acute  bisectrix 420 

Sections  cut  at  right  angles  to  the  obtuse  bisectrix 423 

Sections  inclined  to  the  bisectrices 423 

Sections  at  right  angles  to  an  optic  axis 424 

Sections  parallel  to  the  plane  of  the  optic  axes 425 

Locating  the  point  of  emergence  of  an  optic  axis 425 

Uniaxial  crystals 425 

Biaxial  crystals      426 

CHAPTER  XXX 

OBSERVATIONS  BY  CONVERGENT  POLARIZED  LIGHT  (CONTINUED) 429 

Isotaques,  skiodromes,  and  isogyres 429 

Isotaques  or  curves  of  equal  velocity 429 

Skiodromes 430 

To  construct  the  skiodromes  for  a  random  section 433 

Deduction  of  the  isogyres  from  the  skiodromes 434 

Skiodromes  of  uniaxial  crystals 435 

Sections  cut  at  right  angles  to  the  optic  axis 435 

Sections  inclined  to  the  optic  axis 436 

Sections  parallel  to  the  optic  axis 436 

Skiodromes  of  biaxial  crystals 437 

Sections  perpendicular  to  the  principal  vibration  axes .  437 

Sections  perpendicular  to  the  acute  bisectrix 437 

Sections  perpendicular  to  the  obtuse  bisectrix 437 

Sections  perpendicular  to  the  optic  normal 438 

Sections  perpendicular  to  an  optical  plane  of  symmetry 438 

Sections  perpendicular  to  the  plane  of  the  optic  axes 438 

Inclined  sections 439 

Random  sections 439 

Equations  for  the  isogyres  or  neutral  curves 440 

CHAPTER  XXXI 

DISPERSION  OF  LIGHT  IN  CRYSTALS 442 

Normal  and  anomalous  dispersions 442 

Dispersion  in  orthorhombic  crystals 443 

Dispersion  of  the  optic  axes 443 

Crossed  axial  plane  dispersion 444 


xviii  CONTENTS 

PAGE 

Dispersion  in  monoclinic  crystals  . 445 

Dispersion  of  the  bisectrices 445 

Inclined  dispersion  (of  both  bisectrices) 445 

Horizontal  dispersion  (of  the  acute  bisectrix) 446 

Crossed  dispersion  (of  the  obtuse  bisectrix) 447 

Dispersion  in  triclinic  crystals 448 

Unsymmetrical  dispersion 448 

Effect  of  temperature  change  on  dispersion .  448 

CHAPTER  XXXII 

THE  PETROGRAPHIC  MICROSCOPE  AS  A  CONOSCOPE  AND  THE  METHODS  FOR  OBSERVING 

INTERFERENCE  FIGURES 449 

Observing  interference  figures  with  the  microscope 449 

Lasaulx  method 449 

Bertrand  method  (1878) 449 

Klein  method 450 

Laspeyres  method 451 

Bertrand  method  (1880) 451 

Schroeder  van  der  Kolk  method 452 

Czapski  ocular 453 

Becke-Klein  magnifier      453 

Lenk-Lasaulx  method 453 

Sommerfeldt  condenser 454 

Wright-Lasaulx  method 454 

Johannsen  auxiliary  lens 454 

Orientation  of  image  in  relation  to  object 456 

CHAPTER  XXXIII 

DETERMINATION  OF  THE  OPTICAL  CHARACTER  OF  A  CRYSTAL  BY  MEANS  OF  ITS  INTER- 
FERENCE FIGURE 457 

Positive  and  negative  minerals 457 

Uniaxial  crystals 457 

Quarter  undulation  mica  plate 457 

Unit  retardation  plate      459 

Quartz  or  gypsum  wedge 460 

Inclined  sections 461 

Sections  parallel  to  the  optic  axis 462 

Biaxial  crystals 462 

Mica  plate,  gypsum  plate,  and  quartz  wedge 462 

CHAPTER  XXXIV 

MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  BY  CONVERGENT  POLARIZED  LIGHT  .    .    .  466 

Introduction 466 

Mallard  method  for  sections  showing  the  points  of  emergence  of  both  optic 

axes 467 

Becke's  graphical  solution  of  sin  E  =  n  sin  V 468 

Schwaizmann  axial  angle  scale 469 

Schwarzmann  ocular    .  470 


CONTENTS  xix 

PAGE 

De  Souza-Brandao  axial  angle  diagram 471 

Michel-LeVy  method  for  sections  perpendicular  to  a  bisectrix      472 

Viola  method 474 

Becke  method  for  determining  graphically  the  axial  angle  in  sections  which  do 

not  show  the  points  of  emergence  of  both  optic  axes 476 

Determination  of  the  point  of  emergence  of  an  optic  axis 476 

Becke's  rotating  drawing  stage 478 

Becke  method  for  determining,  by  means  of  the  curvature  of  the  isogyres, 
the  value  of  the  axial  angle  in  sections  which  show  the  point  of  emergence 

of  but  a  single  optic  axis 480 

Wright's  modification  of  the  Becke  method  for  determining  the  axial  angle  by 

means  of  the  curvature  of  the  isogyres 483 

Modifications  of  Becke's  method 485 

Wright 485 

Stark 485 

Tertsch 485 

CHAPTER  XXXV 

MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  BY  MEANS  OF  A  ROTATION  APPARATUS  .    .487 

The  rotation  apparatus 487 

Locating  one  optic  axis 489 

Determination  of  the  position  of  an  optic  axis  by  means  of  the  optical  curves.    .   494 

Locating  the  point  of  emergence  of  the  second  optic  axis 495 

Locating  the  symmetry  planes  and  the  axes  of  the  optical  ellipsoid  within  the 

crystal 497 

Determination  of  the  position  of  the  second  oplic  axis  when  the  first  is  deter- 

minable  by  optical  curves 498 

Approximate  determination  of  the  optic  axes  when  the  section  lies  nearly  parallel 

to  the  plane  of  the  optic  axes 499 

Simplified  methods 500 

Both  optic  axes  appear  in  the  field  of  the  microscope  at  the  most  satisfactory 
angle 500 

One  optic  axis  makes  an  angle  of  less  than  20°  with  the  normal  to  the  section  .  500 

One  optic  axis  makes  an  angle  of  between  20°  and  55°  with  the  normal  to  the 
section,  the  other  lies  beyond  55° 501 

Both  optic  axes  are  inclined  more  than  55°  to  the  normal  to  the  section  .    .   501 

CHAPTER  XXXVI 

DETERMINATION  OF  OTHER  PROPERTIES  THAN  2V  BY  MEANS  OF  THE  UNIVERSAL  STAGE  503 

Opaque  minerals 503 

Isotropic,  uniaxial,  or  biaxial  character 503 

Positive  or  negative  character 503 

Maximum  extinction  angle 504 

Mean  refractive  index  of  a  mineral 504 

Orientation  of  the  crystal  section  with  reference  to  the  axes  of  the  optical 

ellipsoid 504 

Determination  of  the  maximum  birefringence  of  an  unknown  mineral  from 

that  of  one  which  is  known 504 

Graphical  representation  of  the  variation  in  the  double  refraction  in  different 

directions 506 


xx  CONTENTS 

CHAPTER  XXXVII 

PAGE 

OPTICAL  ANOMALIES 508 

The  cause  of  optical  anomalies 508 

CHAPTER  XXXVIII 

'DETERMINATION  OF  SPECIFIC  GRAVITY 515 

Specific  gravity 515 

Hydrostatic  balance 515 

Jolly  balance 516 

Pycnometer  for  determining  the  specific  gravity  of  powders 517 

Smeeth's  method  for  mineral  powders 517 

Specific  gravity  of  porous  substances 518 

Specific  gravity  of  substances  soluble  in  water 518 

Determination  of  specific  gravity  by  heavy  solutions 518 

Sonstadt  (or  Thoulet)  solution 519 

Klein  solution 521 

Rohrbach  solution 524 

Methylene  iodide  (Braun) 525 

Retgers'  heavy  fluids 526 

Tabulation  of  the  properties  of  heavy  fluids 528 

Schroeder  van  der  Kolk 529 

Muthmann 529 

Clerici 529 

Joly 53° 

Hubbard 530 

Streng 530 

Retgers 531 

Behr 531 

Determination  of  the  specific  gravity  of  the  heavy  solution 532 

Goldschmidt's  method 532 

Sprengel  tube 532 

Sollas'  modification  of  the  Sprengel  tube      533 

Westphal  balance 533 

Salomon's  apparatus '. 534 

Sollas'  hydrostatic  float 534 

Merwin's  method  by  refractive  indices 534 

Molten  substances  as  specific  gravity  fluids 535 

Determination  of  the  specific  gravity  of  a  mineral  whose  density  is  greater  than 

that  of  the  fluid 535 

Thoulet 535 

CHAPTER  XXXIX 

THE  MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 537 

Preliminary  examination 537 

Separation  by  means  of  the  electromagnet 538 

Separation  by  means  of  water 541 

Separation  by  means  of  heavy  fluids 542 

Indicators 542 

Table  of  specific  gravities 544 


CONTENTS  xxi 

PAGE 

Heavy  solutions 545 

Heavy  melts 545 

Separating  apparatus 547 

Thoulet 547 

Goldschmidt 548 

Harada 549 

Oebbeke 550 

Van  Werveke 550 

Brogger 551 

Smeeth 552 

Diller 553 

Laspeyres 553 

Wiilfing 553 

Luedecke 554 

Separation  apparatus  for  heavy  melts 554 

Causes  likely  to  produce  errors  in  separating  minerals  or  in  determining  specific 

gravities  by  means  of  heavy  fluids 556 

Separation  of  thin  flakes  and  fine  needles 557 

Separation  by  hand 557 

Separation  by  chemical  means    . 558 

CHAPTER  XL 

MlCROCHEMICAL    REACTIONS '.     .     .  559 

General  microchemical  reactions 559 

Chemical  reactions  on  rock  slices 559 

Apparatus 559 

Preparing  the  slide 560 

Microchemical  nitrations 561 

Gelatinizing  and  staining  minerals 562 

Special  reactions,  chiefly  on  thin  sections 563 

Hauynite,  noselite,  sodalite,  melilite,  and  zeolites 563 

Nephelite,  cancrinite,  and  hydronephelite 564 

Olivine  family 565 

Apatite 565 

Carbonates 565 

Separating  quartz  from  feldspar '. 568 

CHAPTER  XLI 

PREPARATION  OF  THIN  SECTIONS  OF  ROCKS 572 

Early  history 572 

Section  cutting  machines 574 

Diamond  saws 580 

Sawing  a  rock  slice 583 

Grinding  a  section 585 

Various  grinding  machines 588 

Orienting  devices 592 

Mounting  the  section 593 

Special  methods  for  preparing  sections  of  unusual  material 599 

Friable  material ; 599 


xxii  CONTENTS 

PAGE 

Vesicular  rocks 600 

Coal 601 

Clays  and  soft  powders 601 

Sand  and  other  loose  grains 602 

Hydrous  minerals 602 

Minerals  soluble  in  water 602 

The  preparation  of  polished  faces  on  rocks 602 

Rims 602 

CHAPTER  XLII 

PETROGRAPHIC  COLLECTIONS 605 

.Field  work 605 

Working  tools 605 

Hand  specimens 607 

Wrappers  and  labels 608 

Packing  specimens  for  shipment 608 

Office  work 609 

Accession  catalogue 609 

Permanent  labels  for  hand  specimens 609 

Labels  for  thin  sections 610 

Marking  thin  sections 611 

Cases  for  thin  sections 612 

Card  catalogue 613 

APPENDIX 

Greek  alphabet 619 

Useful  formulae 619 

Trigonometric 619 

Cartesian  geometric 621 

Conversion  tables  for  weights  and  measures 623 

Useful  recipes 624 

Table  of  natural  sines  and  cosines 626 

Table  of  natural  tangents  and  cotangents 628 

INDEX 631 


LIST  OF  ABBREVIATIONS1 

Abh.    Akad.   Wiss.   Berlin    =   Abhandlunsjen  der  koniglich  preussischen  Akademie  der 

Wissenschaften,  Berlin.     I  (1770)+. 
Abh.  geol.  Specialkarte  Elsass-Loth.  =  Abhandlungen  zur  geologischen  Specialkarte  von 

Elsass-Lothringen.     Strassburg  i.  E. 
Amer.  Geol.  =  The  American  Geologist.     Minneapolis,  Minn.     I  (i888)-XXXVI  (1905). 

Merged  in  Economic  Geology  in  1906. 
Amer.  Jour.  Microsc.  =  The  American  Journal  of  Microscopy  and  Popular  Science.     New 

York.     I  (i87s)-XII  (1881). 
Amer.  Jour.  Sci.  =  The  American  Journal  of  Science.     New  Haven,  Conn.     I  (1818)+. 

50  volumes  to  a  series. 
Amer.  Mon.  Microsc.  Jour.  =  The  American  Monthly  Microscopical  Journal.     Washing- 

ton, D.  C.     I  (i88o)-XXIII  (1902).     Preceded  by  Amer.  Quart.  Microsc.  Jour. 
Amer.  Nat.  =  The  American  Naturalist.     New  York.     I  (1867)+. 
Amer.  Quart.  Microsc.  Jour.   =  The  American  Quarterly  Microscopical  Journal.     New 

York.     1878-1879.     Continued  as  Amer.  Mon.  Microsc.  Jour. 
Ann.  Chim.  et  Phys.  =  Annales  de  chimie  et  de  physique.     Paris.     I  (1788)+. 
Ann.  d.  Phys.  =  Annalen  der  Physik.    Leipzig.     I  (1799)+.     ist  series,  76  vols.  1799-1824, 

edited  by  L.  W.  Gilbert.     2nd  series,  160  vols.,  1824-1876,  edited  by  J.  C.  Pog- 

gendorff.     3d  series,  69  vols.,  1877-1899,  edited  by  G.  Wiedemann  (Vols.  48-69 

with  E.  Wiedemann).     4th  series,  continued  from  I  (1900)+. 
Ann.  d.  k.  k.  naturhist.  Hofmuseum  =  Annalen  des  k.  k.  naturhistorischen  Hofmuseums. 

Wien.  I(i886)+. 
Ann.  d.  Mines  =  Annales  des  mines.     Paris.     I  (1816)+.     zoth  series  begun  in  1902. 

Continuation  of  Jour.  d.  Mines. 
Anz.  Akad.  Wiss.  Krakau    =    Anzeiger  der  Akademie  der  Wissenschaften  in  Krakau. 

I  (1901)+.     (Akademija  umiejetnosci).     Same  as  Bull.  Acad.  Sci.  Cracovie. 
Arch.  d.  naturwiss.  Landesdurchf.  Bohmen  =  Archiv  der  naturwissenschaftlichen  Landes- 

durchforschung  von  Bohmen.     Prague. 

Arch.  d.  sciences,  physiques  et  natureUe,  see  Bibliotheque  universelle.     Geneve. 
Arch.  f.  Mikrosk.  Anatomic  =  Archiv  fur  mikroskopische  Anatomic  und  Entwicklungs- 

geschichte.     Bonn.     I  (1865)+. 
Arch.    Neer.    =    Archives   neerlandaises    des  sciences  exactes  et  naturelles.     Haarlem. 

I  (1866)+. 
Astron.  and  Astrophys.  =  Astronomy  and  Astrophysics.     Northfield,  Minn.     I  (i882)-XII 

(1894).     Continued  as  The  Astrophysical  Journal.     Chicago.     1(1895)+. 

Ber.  deutsch.  bot.  Gesell.    =    Berichte  der  deutschen  botanischen  Gesellschaft.     Berlin. 


Ber.  deutsch.  chem.  Gesell.   =   Berichte  der  deutschen  chemischen  Gesellschaft.     Berlin. 

I  (1868)+. 
Ber.  Gesell.  Wiss.  Leipzig  =  Berichte  iiber  die  Verhandlungen  der  koniglich  sachsichen 

Gesellschaft  der  Wissenschaften  zu  Leipzig.     I  .(1846)+. 

1  In  most  cases  in  the  following  list,  the  date  of  the  first  volume  issued  is  given  for  bibliographic  in- 
formation. If  a  second  date  appears  it  indicates  that  the  publication  has  been  discontinued.  A  +  sign 
indicates  that  the  series  continues  to  date. 

xxiii 


xxiv  LIST  OF  ABBREVIATIONS 

Ber.  oberhess.  Gesell.  =  Bericht  der  oberhessischen  Gesellschaft  fur  Natur-  und  Heilkunde. 

Giessen.    I  (1847)+. 
Bibliotheque  universelle,  Geneve  =  Originally  Bibliotheque  britannique.     Geneve.     1796- 

1815.     Continued  as  Bibliotheque  universelle  des  sciences,  belles-lettres  et  arts, 

Geneve.     1816-1835.     Continued  further  as  Bibliotheque  universelle  de  Geneve. 

1836-1845.     Now  Bibliotheque  universelle.     Archives  des  sciences,  physiques  et 

naturelles,  Partie  scientifique.     Geneve.     1846+. 
Biol.  Centralbl.  =  Biologisches  Centralblatt.    Leipzig. 
Bot.  Centralbl.  =  Botanisches  Centralblatt.     Cassel. 
Bull.  Acad.  Sci.  Cracovie  =  Bulletin  international  de  1'Academie  des  sciences  de  Cracovie. 

I  (1901).     Same  as  Anz.  Akad.  Wiss.  Krakau. 
Bull.  Acad.  Roy.  Belgique  =  Bulletins  de  1'Academie  royale  des  sciences,  des  lettres  et  des 

beaux-arts  de  Belgique.     Classe  de  Sciences.     Bruxelles.     1(1832)+. 
Bull.  Soc.  Beige  de  Micr.  =  Bulletin  de  la  societe  Beige  de  microscopie.     Bruxelles. 
Bull.  Soc.  Chem.  Paris  =  Bulletin  de  la  societe  chimique  de  France.     Paris.     I  (1858)  +  . 
Bull.   Soc.    Min.   France    =    Bulletin  de  la  societe    francais  de  mineralogie.     Paris.     I 

(1878)+.     Previous  to  1886  Societe  mineralogique  de  France. 

Carl's  Repertorium    =    Repertorium  der  Physik.     Edited  by  Philip   Carl.     Miinchen. 

I  (i866)-XXVII  (1891). 
Centralbl.  f.  Min.,   etc.    =    Centralblatt  fur  Mineralogie,   Geologic  und  Palaontologie. 

Stuttgart.     I  (1900)+. 
Chem.  News  =  The  Chemical  News  and  Journal  of  Physical  Science.    London.     Originally 

The  Chemical  Gazette.     I  (i843)-(i8sg).     Chemical  News.     1860+ . 
Comptes  Rendus    =    Comptes  rendus    hebdomadaires  des  seances  de    1' Academic   des 

Sciences.     Paris.     I  (1835)+. 
Chem.  Zeitschr.  =  Chemische  Zeitschrift.     I  (1901)+. 

Denkschr.  Akad.  Wiss.  Wien  =  Denkschriften  der  mathematisch-natur  wissenschaftliche 

Classe  der  kaiserliche  Akademie  der  Wissenschaften  zu  Wien.     I  (1848)  +  . 
Deutsche  Mechan.  Zeitung  =  Deutsche  Mechaniker  Zeitung.     Berlin.     1(1898)+. 

Econ.  Geol.  =  Economic  Geology,  n.  p.     I  (1905)+. 

Edinburgh  New  Phil.  Jour.  =  The  Edinburgh  New  Philosophical  Journal.  Edinburgh 
(1826-1864).  Originally  Edinburgh  Philosophical  Journal  (1819-1825).  The 
Edinburgh  New  Philosophical  Journal  (1826-1854).  New  Series  (1855- 
1864).  Merged  in  The  Quarterly  Journal  of  Science,  1864. 

English  Mechanic  =  English  Mechanic  and  World  of  Science.     London.     I  (1865)+. 

Foldtani  Kozlony  =  Foldtani  Kozlony  (Geological  Communications).     Budapest. 
Fortschritte  der  Min.,  Kryst.,  und  Petrog.   =   Fortschritte  der  Mineralogie,  Kristallo- 
graphie  und  Petrographie.     Jena.     1(1911)+. 

Gilbert's  Ann.  =  Gilbert's  Annalen  der  Physik.     See  Ann.  d.  Phys. 

Geol.  Foren.  i  Stockholm   Forh.   =   Geologiska   foreningens  i   Stockholm  forhandlingar. 

Stockholm.     I  (1872)+. 
Geol.  Mag.  =  The  Geological  Magazine.    London.     Originally  The  Geologist.     I  (1858- 

1863).     The  Geological 'Magazine,  I  (1864)+. 
Grunert's  Arch.  =  Archiv  der  Mathematik  und  Physik.  Leipzig  und  Berlin.     I  (1841)+. 

Founded  by  J.  Grunert. 


LIST  OF  ABBREVIATIONS  xxv 

Jahresh.  d.  Ver.  f.  vaterl.  Naturk.  Wurttemberg  =  Jahreshefte  des  Vereins  fur  vater- 

landische  Naturkunde  in  Wurttemberg. 
Jour,  and  Proc.  Roy.  Soc.  New  So.  Wales  =  Journal  and  Proceedings  of  the  Royal  Society 

of  New  South  Wales.     Sydney. 
Jour.    Appl.    Microsc.    =    Journal   of   Applied   Microscopy   and  Laboratory   Methods. 

Rochester,  N.  Y.     I  (i898)-VI  (1903). 

Jour.  Chem.  Soc.  London  =  The  Journal  of  the  Chemical  Society.    London.     I  (1849)4-. 
Jour.  d.  Mines  =  Journal  des  mines  (1795-1815).     Continued  as  Ann.  d.  Mines,  1816+. 
Jour.  d.  Phys.  =  Journal  de  physique  theoretique  et  appliquee.     Paris.     I  (1872)  +  . 
Jour.  Geol.  =  The  Journal  of  Geology.     Chicago.     I  (1893)+. 
Jour.  Microsc.   =   The  Journal  of  Microscopy  and  Natural  Science.     See  Jour.  Postal 

Microsc.  Soc. 
Jour.   N.   Y.   Microsc.    Soc.   =  The  Journal  of   the  New  York  Microscopical  Society. 

I  (i88s)-XIV  (1898). 
Jour.  Postal  Microsc.  Soc.  =  The  Journal  of  the  Postal  Microscopical  Society.    London. 

I   (1882)-!!  (1883).     Succeeded  by  The  Journal  of  Microscopy  and  Natural 

Science.     Ill  (i883)-XVI  (1897). 
Jour.  Roy.  Microsc.  Soc.   =  The  Journal  of  the  Royal  Microscopical  Society.    London. 

Preceded  by  Transactions  of  the  Mineralogical  Society  (1844-1868),  The  Monthly 

Microscopical  Journal   (1860-1877).  The  Journal  of  the  Royal  Microscopical 

Society,  1878+. 

Jour.  Roy.  Soc.  Arts  =  The  Journal  of  the  Royal  Society  of  Arts.    London.     I  (1852)+. 
Jour.  Washington  Acad.  Sci.   =  The   Journal  of  the  Washington  Academy  of  Science. 

Washington,  D.  C.     I  (1911)+. 

Knowledge  =  Knowledge.     London.     I  (1881)+ . 

Mem.  Acad.  France    =    Memoires  de  1'Academie  des  sciences  de  ITnstitut  de  France. 

I  (1796)+.     Various  slight  variations  in  the  title. 
Mem.  Acad.  Sci.  Belgique   =   Memoires  couronnes  et  memoires    des  savants  etrangers 

publies  par  1' Academic  royale  des  sciences,  des  lettres  et  des  beaux-arts  de 

Belgique.     I  (i8i7)-LXII  (1904).     Bruxelles. 

Memoires   couronnes   et  autres   memoires  publies   par  rAcademie  royale  des 

sciences,  des  lettres  et  des  beaux-arts  de  Belgique.     Collection  in  8vo.     I  (1840)- 

LXVI  (1904).     Bruxelles. 

Beginning  with  1906  all  the  Memoirs  of  the  Academy  are  published  in  two  series. 

A,  sciences,  B,  Lettres,  sciences  morales  et  politiques.     Each  series  includes  two 

collections,  one  in  4to  and  one  in  8vo. 
Mem.  Accad.  Sci.  Napoli  =  Memorie  delPaccademia  delle  scienze  fisiche  e  matematiche. 

Napoli. 
Mem.  and  Proc.  Chem.  Soc.,  London  =  Memoirs  and  Proceedings  of  the  Chemical  Society 

of  London.     I  (i84i)-III  (1848).     Continued  as  Jour.  Chem.  Soc.  London. 
Microsc.  Bull.  =  The  Microscopical  Bulletin  and  Science  News.     Philadelphia.     I  (1883)- 

XVIII  (1901). 

Microsc.  News  =  The  Microscopical  News.     See  Northern  Microsc. 
Microscope  =  The  Microscope.     Washington,  D.  C.  and  v.  p.     I  (i88i)-V  (1897). 
Mineralog.  Mag.  =  The  Mineralogical  Magazine  and  Journal  of  the  Mineralogical  Society 

of  Great  Britain  and  Ireland.     I  (1876)+. 
Mon.  Microsc.  Jour.   =  The  Monthly  Microscopical  Journal.    London.     Continued  as 

Jour.  Roy.  Microsc.  Soc.,  quod  vide. 
Morphol.  Jahrb.  =  Morphologisches  Jahrbuch.     Leipzig. 


xxvi  LIST  OF  ABBREVIATIONS 

Nature  =  Nature,  a  Weekly  Illustrated  Journal  of  Science.    London.     I  (1869)+. 

Nachr.  Gesell.  Wiss.  Gottingen  =  Nachrichten  der  kgl.  Gesellschaft  der  Wissenschaften 
zu  Gottingen. 

National  Druggist  =  National  Druggist.     St.  Louis. 

Neues  Jahrb.  =  Originally  Leonhard's  Taschenbuch  fur  die  Gesammte  Mineralogie, 
Frankfurt  a.M.  (1807-1824),  Leonhard's  Zeitschrift  fur  Mineralogie  (1825-1829), 
Leonhard  und  Bronn's  Jahrbuch  fur  Mineralogie,  Geognosie,  Geologic,  undPetre- 
faktenkunde,  Heidelberg  (1830-1832),  Neues  Jahrbuch  fur  Mineralogie,  Geog- 
nosie, Geologie,  und  Petrefaktenkunde,  Heidelberg  (1833-1862),  Neues  Jahrbuch 
fur  Mineralogie,  Geologic,  und  Palaeontologie.  Stuttgart.  (1879)+. 

Neues  Jahrb.,  B.B.  =  Neues  Jahrbuch,  etc.,  Beilage  Band.     I  (1883)+. 

Nicholson's  Journal  =  A  Journal  of  Natural  Philosophy,  Chemistry,  and  the  Arts.  Lon- 
don. I  (i7Q7)-V  (1801),  N.  S.  I  (i8o2)-XXXVI  (1813). 

Northern  Microsc.  =  The  Northern  Microscopist.  London.  (1881.)  Followed  by  The 
Microscopical  News  and  Northern  Microscopist.  London.  1882-1883. 

Notizbl.  Ver.  Erdk.  Darmstadt  =  Notizblatt  des  Vereins  fiir  Erdkunde  zu  Darmstadt  und 
des  mittelrheinischen  geologischen  Vereins.  Darmstadt.  I  (1858)+. 

Phil.  Mag.  =  The  Philosophical  Magazine.  London.  1798-1832.  United  in  1832  with 
the  Edinburgh  Journal  of  Science  under  the  title  London  and  Edinburgh  Philo- 
sophical Magazine  and  Journal  of  Science,  I  (i832)-(i85o),  followed  by  Lon- 
don, Edinburgh  and  Dublin  Philosophical  Magazine  and  Journal  of  Science, 
I  (1851)+. 

Phil.  Trans.  Roy.  Soc.  London  =  The  Philosophical  Transactions  of  the  Royal  Society  of 
London.  I  (1665)+. 

Physical  Review  =  The  Physical  Review.     New  York.     I  (1894)+. 

Pogg.  Ann.  =  See  Ann.  der  Phys. 

Proc.  Amer.  Acad.  =  Proceedings  of  the  American  Academy  of  Arts  and  Sciences.  Boston. 
I  (1846)+. 

Proc.  Amer.  Microsc.  Soc.  =  Proceedings  of  the  American  Microscopical  Society,  v.p. 
Originally  Transactions  of  the  American  Microscopical  Society  (1878).  Proceed- 
ings of  the  National  Microscopical  Congress,  Vols.  I  to  II;  Proceedings  of  the 
American  Society  of  Microscopists,  Vols.  II  to  XIV;  Proceedings  of  the  American 
Microscopical  Society,  XV  to  XVII. 

Proc.  Cambridge  Phil.  Soc.  =  Proceedings  of  the  Cambridge  Philosophical  Society. 
Cambridge  (England).  I  (1843)+. 

Proc.  Liverpool  Geol.  Asso.  =  Proceedings  of  the  Liverpool  Geological  Association.  Liver- 
pool. I  (1860)+. 

Proc.  Rochester  Acad.  Sci.  =  Proceedings  of  the  Rochester  Academy  of  Science. 
Rochester,  N.Y.  I  (1889)+. 

Proc.  Geol.  Soc.  London  =  Proceedings  of  the  Geological  Society  of  London.  1826-1845. 
Continued  in  Quart.  Jour.  Geol.  Soc.,  London.  1845+. 

Proc.  Roy.  Soc.  Edinburgh  =  Proceedings  of  the  Royal  Society  of  Edinburgh. 
I  (i84S)+. 

Proc.  Roy.  Soc.  Dublin  =  Scientific  Proceedings  of  the  Royal  Dublin  Society.     I  (1856)  +. 

Proc.  Roy.  Soc.  London  =  Proceedings  of  the  Royal  Society  of  London.     I  (1800)  +  . 

Proc.  Roy.  Soc.  Victoria  =  Proceedings  of  the  Royal  Society  of  Victoria.  Melbourne. 
I  (1897)+- 

Prometheus  =  Prometheus.  Illustrirte  Wochenschrift  iiber  die  Fortschritte  in  Gewerbe, 
Industrie  und  Wissenschaft.  I  (1889)+. 


LIST  OF  ABBREVIATIONS  xxvii 

Quart.  Jour.  Geol.  Soc.  London  =  The  Quarterly  Journal  of  the  Geological  Society  of 

London.     I  (1845)+. 
Quart.  Jour.  Microsc.  Sci.   =   The  Quarterly  Journal  of  Microscopical  Science.    London. 

I  (1853)+- 

Rend.  Accad.  Napoli.  =  Rendiconto  delTAccademia  delle  Scienze  Fisiche  e  Mathematiche. 

Napolio. 
Rend.   Accad.  Lincei,   Roma    =    Rendiconti  della  Reale  Accademia  dei  Lincei,  Roma. 

I  (1840  ?)+•     Continuation  of  Atti  and  Transunti  della,  etc. 
Rep.  Brit.  Asso.  Adv.  Sci.=  Report  of  the  British  Association  for  the  Advancement  of 

Science.    London.     I  (1831)+. 
Rivista  di  Min.  Crist.  Ital.  =  Rivista  di  Mineralogia  e  Cristallografia  Italiana.     Padua. 

Vol.  XVIII  is  1897+. 

Schlomilch's  Zeitschr.   =  Zeitschrift  fur  Mathematik  und  Physik.     Founded  in  1856  by 

O.  Schlomilch.    Leipzig.     I  (1856)+. 
Science  =  Science.     New  York.     I  (1880)+. 
Sci.  Gossip  =  Science  Gossip.    London.     I  (i865)-(i902). 
Sitzb.  Akad.  Wiss.  Berlin  =  Sitzungsberichte  der  koniglich  preussischen  Akademie  der 

Wissenschaften.     Berlin.     I  (1836)+. 

Sitzb.  Akad.  Wiss.  Heidelberg  =  Sitzungsberichte  der  Heidelbergei  Akademie  der  Wissen- 
schaften. 
Sitzb.  Akad.  Wiss.  Miinchen.  =  Sitzungsberichte  der  koniglich  Bayerischen  Akademie  der 

Wissenschaften  zu  Munchen.     Vol.  I  (1860)+.     Since  1870  the  Math.-Phys.  Cl. 

and  the  Phil.-Histor.  Cl.  publish  separate  Sitzungsberichte. 
Sitzb.  Akad.  Wiss.  Wien  =    Sitzungsberichte  der  mathematisch-naturwissenschaftlichen 

Klasse  der  kaiserlichen  Akademie  der  Wissenschaften.     Wien.     I  (1848)+. 
Sitzb.  Gesell.  Wiss.  Prag.  =  Sitzungsberichte  der  mathematisch-naturwissenschaftlichen 

Classe  der  koniglich  bohmischen  Gesellschaft  der  Wissenschaften.     Prag. 
Sitzb.  niederrhein.  Gesell.  Bonn   =   Sitzungsberichte  der  niederrheinischen  Gesellschaft 

fur  Natur-  und  Heilkunde  zu  Bonn.     1854-1906.     Continued  in  Sitzungsberichte 

herausgegeben  von  Naturhistorischen  Verein  der  preussischen  Rheinlande  und 

Westfalens. 

Trans.  Amer.  Inst.  Mining  Eng.   =  Transactions  of  the  American  Institute  of  Mining 

Engineers.     New  York.     1(1871)+. 

Trans.  Liverpool  Geol.  Asso.  =  Transactions  of  the  Liverpool  Geological  Association. 
Trans.  Roy.  Irish   Acad.    =    Transactions   of   the   Royal  Irish  Academy.     Dublin.     I 

(1787)+. 
Trans.    Roy.    Soc.    Edinburgh    =    Transactions    of    the   Royal    Society  of  Edinburgh. 

I  (1783)+- 

T.  M.  P.  M.  =  Tschermak's  Mineralogische  und  Petrographische  Mitteilungen,  Vienna. 
Originally  Mineralogische  Mittheilungen,  1871-1877,  continued  as  above. 

I  (1878)+. 

U.  S.  G.  S.,  Ann.  Rept.  =  Annual  Report  of  the  United  States  Geological  Survey.  Wash- 
ington, D.  C.  I  (1880)+. 

U.  S.  G.  S.,  Bull.  =  Bulletin  of  the  United  States  Geological  Survey.  Washington,  D.  C. 
No.  I  (1883)+. 

U.  S.  G.  S.,  Mono.  =  Monograph  of  the  United  States  Geological  Survey.  Washington, 
D.  C.  I  (1890)+. 

U.  S.  G.  S.,  P.  P.  =  Professional  Paper  of  the  United  States  Geological  Survey.  Wash- 
ington, D.  C.  No.  I  (1902)+. 


xxviii  LIST  OF  ABBREVIATIONS 

Versl.  en  Meded.  Akad.  Weten.  Amsterdam  =  Verslagen  en  Mededeelingen  der  Koninklijke 

Akademie  van  Wetenschappen  te  Amsterdam.    Afdeeling  natuurkunde.    I  (1855)- 

IX  (1892). 
Verb.  k.  k.  Geol.  Reichsanst.  Wien.  =  Verhandlungen  der  k.  k.  geologischen  Reichsanstalt. 

Wien.     I  (1867)+. 
Verb.   Russ.  Min.   Gesell,   St.   Petersburgh    =    Verhandlungen  der  russisch-kaiserlichen 

Mineralogischen  Gesellschaft  zu  St.  Petersburgh. 
1  Verb.  Phys.  Med.  Gesell.  Wiirzburg    =    Verhandlungen  der  physikalisch-medicinischen 

GeseUschaft  zu  Wurzburg.     I  (1850)  +  . 
Verb.  Naturf.  Gesell.  Basel  =   Verhandlungen  der  naturforschende  Gesellschaft.     Basel. 

Vol.  VII  (1885)+. 
Verb.  Naturhist.  Ver.  Preuss.  Rheinl.  Bonn.    =   Verhandlungen  des  naturhistorischen 

Vereins  der  preussischen  Rheinlande  und  Westfalens.     Bonn.     1(1844)+. 

Wiedem.  Ann.  Weidemann's  Annalen.     See  Ann.  der  Phys. 

Zeitschr.  f.  analyt.  Chemie.  =  Zeitschrift  fur  analytische  Chemie.     Wiesbaden.     I  (1862)  +. 
Zeitschr.  f.  angew.  Mikrosk.  =  Zeitschrift  fur  angewandte  Mikroskopie,  u.  s.  w.     Berlin, 

Leipzig  und  Weimar.     I  (1895)+. 
Zeitschr.  d.  deutsch.  geol.  Gesell.   =  Zeitschrift  der  deutschen  geologischen  Gesellschaft. 

Berlin.     I  (1849)+. 

Zeitschr.  f.  Instrum.  =  Zeitschrift  fur  Instrumentenkunde.     Berlin.     I  (1881)+. 
Zeitschr.    f.    Kryst.     =    Zeitschrift    fur    Krystallographie    und    Mineralogie.    Leipzig. 

I  (1877)+- 
Zeitschr.   f.   wiss.   Mikrosk.    =    Zeitschrift  fur  wissenschaftliche  Mikroskopie  und    fur 

mikroskopische  Technik.     Leipzig.     I  (1884)  +  . 
Zeitschr.  f.  gesammten  Naturwiss.  =  Zeitschrift  fiir  die  gesammten  Naturwissenschaften. 

Halle.     I  (1853)+. 
Zeitschr.  f.  physik.  Chemie  =  Zeitschrift   fur  physikalische  Chemie,  Stochiometrie  und 

Verwandtschaftslehre.     Leipzig.     I  (1887)+. 


MANUAL  OF  PETROGRAPHIC  METHODS 

CHAPTER  I 
MINERALOGICAL  PRINCIPLES 

1.  Crystals. — Minerals   may   occur   either   crystallized   or   amorphous. 
When  crystallized,  they  possess  certain  properties  which  are  alike  in  parallel 
directions;  when  amorphous,  the  properties  show  no  regular  or  uniform  varia- 
tions.    Substances  which  crystallize,  when  left  free  to  grow  as  they  will, 
tend  to  assume  definite  forms  which  are  characteristic  for  that  mineral. 

Not  only  do  crystals  tend  to  build  up  regular  forms,  but  there  is  a  definite 
molecular  arrangement  throughout  their  mass,  so  that,  as -we  shall  see,  we 
are  enabled,  by  certain  optical  examinations,  to  determine  their  character- 
istics regardless  of  accidental,  favorable  conditions  of  growth. 

2.  Crystallographic  Axes. — The  faces  which   develop  upon   a  crystal 
may  be  referred  to  certain  imaginary  axes,  generally  regularly  arranged, 
always,  however,  having  a  definite  position  in  a  given  mineral.     In  general 
these  axes  are  three  in  number,  and  the  various  faces  may  be  defined  by  their 
intercepts  upon  them.     According  to  the  kinds  of  axes,  we  may  divide  all 
crystals  into  six  (or  seven)  groups.     Without  going  into  the  question  of 
symmetry,  it  is  simplest  to  describe  the  different  systems  in  the  order  of  de- 
creasing complexity. 

I.  In  the  isometric1  system  the  faces  are  referred  to  three  interchangeable 
axes  at  right  angles  to  each  other.     In  ideal  crystals  and  in  drawings,  these 
axes  are  represented  as  of  equal  lengths;  in  nature  they  are  usually  not  alike. 
It  is  customary  to  consider  one  axis  (c)  vertical,  one  extending  from  left  to 
right  (b),  and  one  from  front  to  back  (a).     The  angles  between  these  axes 
are  expressed  by  a  for  that  between  c  and  b,  by  /?  for  that  between  c  and  a, 
and  by  f  for  that  between  a  and  b.     In  this  system  they  are  all  90°  (Fig.  i). 

II.  In  the  tetragonal2  system  the  faces  are  referred  to  three  axes  at  right 
angles  to  each  other,  two  of  them  being  interchangeable,  the  other  either 
longer  or  shorter.     The  two  equal  axes  are  the  a  and  b,  the  unequal  axis  is 
the  c.     a  =  b^c,  a  =  p  =  r  =  go°; 

III.  In  the  hexagonal3  system  there  are  four  axes.     The  three  horizontal 
axes  are  interchangeable  and  inclined  60°  to  each  other;  the  vertical  one  (c) 

1  Tessular,  Mohs;  Isometric,  Hausmann;   Tesseral,  Xaumann;  Regular,  Weiss,  Rose; 
Cubic,  Dufrenoy,  Miller,  des  Cloizeaux;  Monometric,  Dana's  original  system. 

2  Pyramidal,  Mohs;  Viergliedrige  oder  Zwei-und-einaxige,  Weiss;  Tetragonal,  Naumann; 
Monodimetric,  Hausmann;  Quadratic,  von  Kobell;  Dimetric,  Dana  originally. 

3  Rhombohedral,  Mohs;  Sechsgliedrige  oder  Drei-und-einaxige,  Weiss;  Hexagonal,  Xau- 
mann; Monofrimetric,  Hausmann. 

1 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  2 


is  at  right  angles  to  the  plane  of  the  other  three,  and  is  either  longer  or  shorter. 
One  of  the  short  axes  (az)  is  conventionally  considered  as  extending  from  left 
to  right.  The  intercepts  are  written  in  the  order  a\,  az,  as,  c  (Fig.  2). 


03 


-ttj 


-a 


-a3 


-a, 

FIG.   i.  FIG.  2.  FIG.  3. 

FIG.  I. — The  crystallographic  axes  in  the  isometric  system.     a  =  b  =  c,  a  =  0=^  =  90°. 
FIG.  2. — The  crystallographic  axes  in  the  hexagonal  system.     01  =  02  =  03  ^c. 
FIG.  3.: — The  crystallographic  axes  in  the  trigonal  system  when  referred  to  three  axes.     a  =  b  —  c. 

Ilia.  The  trigonal1  system  is  sometimes  considered  as  independent  of 
the  preceding,  and  includes  its  hemimorphic  forms.  It  is  usually  referred  to 
four  axes  arranged  as  in  the  hexagonal  system.  Originally,  however,  it  was 


—b 


-a 


FIG.  4.  FIG.  5. 

FIGS.  4  TO  6. — The  crystallographic  axes  in  the  monoclinic  system. 


-c 


FIG.  6. 


referred  by  Miller  to  three,  and  this  method  is  still  followed  occasionally. 
The  axes  are  interchangeable  and  oblique  (Fig.  3). 

IV.  In  the  orthorhombic2  system  the  faces  are  referred  to  three  unequal 
axes  at  right  angles  to  each  other,     a^b^c,  a  =  /?  =  7*  =  90°. 

V.  In  the  monoclinic3  system  the  faces  are  referred  to  three  unequal 

1  Trigonal,  Groth. 

2  Prismatic   or  Orthotype,   Mohs;   Ein-und-einaxige,   Weiss;   Rhombic  or  Anisomeiric, 
Naumann;  Trimetric  or  Orthorhombic,  Hausmann;  Trimetric,  Dana  originally. 

3  Hemi- prismatic  and  Hemi-orthotype,  Mohs;  Zwei-und-eingliedrige,  Weiss;  Monoclino- 
hedral,  Naumann;  Clinorhombic,  von  Kobell,  Hausmann,  des  Cloizeaux;  Augitic,  Haidinger; 
Oblique,  Miller;  Mono  symmetric,  Groth. 


ART.  5]  MINERALOGICAL  PRINCIPLES  3 

axes,  one  of  which  (a)  is  inclined  in  the  plane  of  the  vertical  axis  (c) ;  the  other 
two  (c,  b)  are  at  right  angles  to  each  other.  The  inclined  axis  (a)  projects 
downward  from  back  to  front,  and  the  acute  angle  between  it  and  c  is  called 
,.?.  a^b^c,  a  =  r  =  po°,  /?<9o°  (Figs.  4-6). 

VI.  In  the  triclinic1  system  there  are  three  unequal  axes,  none  of  which 
is  at  right  angles  to  any  other,     a^b^c. 

3.  The  Weiss  Parameters. — It  was  stated  above  that  the  various  crys- 
tallographic  forms  are  denned  by  their  intercepts  upon  the  axes.     These 
parameters,  as  the  intercepts  are  called,  have  been  variously  expressed  by 
different  writers,  but  at  the  present  time  only  three  systems  are  more  or  less 
used.     The  first  of  these  is  that  of  Weiss,2  who  denoted  the  semi-crystallo- 
graphic  axes  by  the  letters  a,  b,  and  c,  and  indicated  the  position  of  any  face 
by  the  ratio  of  its  intercepts  upon  them.     For  example,  ia:ib  :  20  indicates 
that  the  face  cuts  the  a  and  b  axes  at  unity  and  the  c  at  twice  that  distance, 
id :  2b  :ic  indicates  that  it  cuts  the  a  and  c  axes  at  unity  and  the  b  at  twice 
that  distance,  and  i  a  :  °°  b  :  2C  indicate  that  it  cuts  the  a  axis  at  unity, 
the  b  at  infinity — that  is,  it  is  parallel  to  b — ,  and  the  c  at  twice  unity. 

4.  The  Naumann  System. — The  Weiss  system  was  simplified  by  Naumann  ? 
who  omitted  the  designation  of  the  axes,  wrote  the  intercepts  in  inverse 
order — that  is  c,  b,  a — ,  made  one  of  the  axes,  usually  a,  unity  and  omitted 
writing  it,  and  inserted,  after  the  number  referring  to  the  c  axis,  the  letter  O 
in  the  isometric  system  and  the  letter  P  in  the  others.     By  his  method  the 
three  forms  given  above  become,  2Pt  P2}  and  2P&>. 

5.  The  Miller  Indices. — The  two  preceding  systems  have  been  gradually 
superseded  by  the  so-called  Miller  system.     This  is  the  one  in  common  use 
at  the  present  time  and  is  the  one  used  in  this  book.     It  was  proposed  by 
Whewell4  in   1825  and  soon  after,  independently,  by  Grassmann5  and  by 
Frankenheim.6    It  did  not  come  into  common  use,  however,  until  Professor 

1  Tetar to- prismatic,    Mohs;    Ein-und-eingliedrige,    Weiss;    Triclinohedral,    Naumann; 
Clinorhomboidal,  von  Kobell;  Anorthic,  Haidinger,  Miller;  Anorthic  or  doubly  oblique,  des 
Cloizeaux;  Asymmetric,  Groth. 

2  C.    S.    Weiss:  Krystallographische    Fundamentalbestimmung    des  Feldspathes.     Abh. 
Akad.  Wiss.  Berlin,  Physik.  Kl,  1816-17,  231-285,  especially  footnote  p.  244. 

Idem:  Ueber  eine  verbesserte  Methode  fur  die  Bezeichnung  der  verschiedenen  Fldchen 
eines  Krystallisationssystems.     Ibidem,  1816-17,  286-336. 

3  Carl  Fr.  Xaumann:  Grundriss  der  Krystallographie.    Leipzig,  1826. 

4  W.  Whewell:  A  general  method  of  calculating  the  angles  made  by  any  planes  of  crystals 
and  the  lau'S  according  to  U'hich  they  are  formed.     Phil.  Trans.  Roy.  Soc.  London,  Pt.  I  (1825), 
87-130. 

5  J.  G.  Grassmann:  Zur    physischen  Krystallonomie  und  geometrischen  Combinations- 
lehre,  Stettin,  1829.* 

fi  M.  L.  Frankenheim:  De  crystallorum  cohaesione.     Vratislaviae,  1829.* 


4  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  6 

Miller1  of  Cambridge  adopted  it  in  his  writings,  more  especially  in  his  Crys- 
tallography. In  this  system  the  intercepts  are  written  in  the  order  a,  b,  c 
and  are  expressed  as  reciprocals  of  the  values  given  in  the  Weiss  system. 
That  is,  the  Weiss  parameters  may  be  obtained  by  considering  the  Miller 
indices  as  the  denominators  of  fractions  and  then  reducing  them  to  whole 
numbers.  The  Weiss  forms  a  :b  :  2c,  a  :  2b  :  c,  a  :  cob  :  2c,  a  :  &>b  :  °°c,  be- 
come (221),  (212),  (201),  and  (100).  Intercepts  on  the  negative  ends  of  the 
axes  are  written  with  a  minus  sign  above  the  figures  thus,  ill,  122,  etc. 

6.  Zones. — When  the  intersections  of  certain  faces  are  mutually  parallel, 
and  consequently  parallel  to  the  same  line — called  the  zone-axis — drawn 
through  the  intersection  of  the  crystallographic  axes,  the  faces  are  said  to  lie 
in  a  zone.  Examples  of  zones  are  100,  101,  ooi,  101,  100,  101,  ooi,  101;  and 
ooi,  oio,  ooi,  oio. 

1  W.  H.  Miller:  On  the  forms  of  sulphur  el  of  nickel  and  other  substances.  Phil.  Mag., 
VI  (1835),  104-107. 

Idem:  Ueber  die  Krystallform  des  Schwefelnickels  und  anderer  Substanzen.  Pogg. 
Ann.,  XXXVI  (1835),  475~479- 

Idem:  A  treatise  on  crystallography.     Cambridge,  1839.* 

See  also  E.  von  Fedorow:  Die  Miller schen  sind  die  allein  zulassigen  Symbole.  Zeitschr. 
f.  Kryst.,  XXIV  (1894-5),  132-136. 


CHAPTER  II 
STEREOGRAPHIC  PROJECTION1 

7.  Introductory. — If  a  crystal  be  placed  in  the  center  of  a  sphere,  and 
any  point  upon  it  be  connected  by  a  straight  line  with  the  center  and  the 
surface,  this  point  will  be  definitely  located  by  the  latter  intersection.     Since 
it  is  generally  not  practicable  to  use  a  sphere  to  show  crystal  properties, 
various  methods  of  projection  upon  a  flat  surface  have  been  devised.     Among 
these  the  most  common  are  orthographic,  gnomonic,2  and  stereographic 
projections;  and,  of  these,  the  latter  is  the  one  which  has  been  found  most 
convenient  in  crystallography. 

The  method  of  representing  the  surface  of  a  globe  in  stereographic  pro- 
jection appears  to  have  been  invented  by  the  astronomer  Hipparchus  about 
the  middle  of  the  second  century  before  Christ.  It  was  used  by  Ptolemy 
about  three  hundred  years  later  in  map  making,  and  has  been  used,  more  or 
less,  until  the  present  time.  In  crystallography  it  was  first  used  by  Neu- 
mann,3 whose  book  does  not  appear  to  have  been  appreciated,  however,  for 
only  the  first  part  was  issued.  The  method  was  later  quite  extensively  used 
by  Miller4  in  his  Crystallography. 

8.  Definitions. — In  the  following  discussion  it  will   be  convenient  to  use 
certain  terms  with  definite  meanings.     If  we  consider  the  line  connecting  the 
north  and  south  poles  of  a  sphere  as  vertical,  the  north  pole  will  be  uppermost 
in  the  projection,  and  .we  may  say: 

A  great  circle  is  one  whose  plane  passes  through  the  center  of  the  sphere. 
It  is  the  largest  circle  that  can  be  described  upon  it. 
A  small  circle  is  any  circle  less  than  a  great  circle. 

1  See  general  bibliography  at  end  of  chapter. 

2  For  gnomonic  projection  see: 

E.  Mallard:  Traite  de  cristallographie.     Paris,  1879,  I,  63-66.* 

H.  A.  Miers:  The  gnomonic  projection.     Mineralog.  Mag.,  VII  (1887),  145-149. 

V.  Goldschmidt:  Projection  und  graphische  Krystallberechnung.     Berlin,  1887.* 

N.  Story-Maskelyne:  Crystallography.  A  treatise  on  the  morphology  of  crystals.  Oxford, 
1895,  492-499.* 

G.  F.  Herbert  Smith:  On  the  advantages  of  the  gnomonic  projection  and  its  use  in  the 
drawing  of  crystals.  Mineralog.  Mag.,  XIII  (1913),  309-321. 

Idem:  Ueber  die  Vorziige  der  gnomonischen  Projection  und  iiber  ihre  Anwendung  beini 
Krystalheichnen.  Zeitschr.  f.  Kryst.,  XXXIX  (1903-4),  142-152. 

Harold  Hilton:  The  gnomonic  net.     Mineralog.  Mag.,  XIV  (1904),  18-20. 

H.  E.  Boeke:  Die  gnomonische  Projektion  in  ihrer  Anwendung  anf  kristallo  graphic  he 
Aufgabcn.  Berlin,  1913. 

3  F.  Neumann:  Bcitrage  zur  Krystallonomie.     Berlin  und  Posen,  1823.* 

4  W.  H.  Miller:  A  treatise  on  crystallography.     Cambridge,  1839.* 

Idem:  On  the  employment  of  the  stereographic  projection  of  the  sphere  in  crystallography. 
Phil.  Mag.,  XIX  (1860),  325-328. 

5 


MANUAL  OF  PETROGRAPIIIC  METHODS 


[ART.  9 


Vertical  great  circles  are  those  which  pass  through  the  north  and  south 
poles.  Their  projections  are  straight  lines,  and  their  centers  lie  in  the  equa- 
torial plane.  These  lines  may  be  called  meridians. 

Vertical  small  circles  are  circles  whose  centers  lie  on  the  equator  and  whose 
radii  are  less  than  90°.  They  are  projected  as  circles. 

The  horizontal  great  circle  is  the  equator. 

Horizontal  small  circles  correspond  to  parallels  of  latitude  and,  conse- 
quently, may  be  called  parallels. 

Antipodal  points  are  points  on  the  sphere  at  opposite  ends  of  lines  passing 
through  the  center.  Thus  the  north  and  south  poles  are  antipodal  points. 

The  pole  of  a  face  is  the  point  where  a  line,  drawn  at  right  angles  to  the 
face  and  passing  through  the  center  of  the  sphere,  pierces  the  latter.  The 
term  is  also  applied  to  the  stereographic  projection  of  this  point. 

9.  Locating  Points. — In  making  a  stereographic  projection,  all  lines  and , 
points  of  a  crystal  must  first  be  imagined  as  projected  upon  a  sphere 


no 


010 


FIG.  7. — Perspective  view  of  a  sphere  sur- 
rounding a  crystal  of  diopside,  showing  the  loca- 
tion of  the  poles,  etc. 


110 


FIG.  8. — Stereographic  projection  of  the  same. 


(Fig.  7),  crystal  faces  being  represented  by  the  piercing  points  of  lines  ex- 
tending at  right  angles  to  them  and  through  the  center  of  the  sphere.  If, 
now,  the  eye  be  placed  at  the  south  pole,  all  of  these  lines  and  points  can 
be  traced  upon  a  transparent  plane  lying  in  the  plane  of  the  equator  (Fig.  8). 
Let  the  circle  in  Fig.  9  represent  a  north  and  south  section  through  a 
sphere  along  a  meridian.  The  eye  being  placed  at  the  point  S,  we  will 
observe  the  intersections  of  the  meridian  with  the  10°,  20°,  30°,  etc.,  parallels, 
as  points  upon  the  line  WE,  the  intersections  in  the  southern  hemisphere 
being  represented  by  points  beyond  the  circle.  Seen  in  stereographic  pro- 
jection, these  points  will  appear  at  a,  b,  c,  etc.,  as  shown  in  Fig.  10  on  the 
line  /'/.  On  some  other  meridian  the  intersections  of  the  same  parallels 
will  appear  at  the  points  a',  b',  c',  etc.  The  distances  of  these  points  from  the 
center  are  measured  by  the  tangents  of  half  the  angles  made  by  the  lines 
from  the  south  pole  through  them,  the  radius  being  taken  as  unity,  for,  by 
geometry,  the  angle  BSN=i/2  BON  (Fig.  n).  Since  BON  is  measured  by 


ART.  9] 


STEREOGRAPHIC  PROJECTION 


the  arc  NB,  BSN  is  equal  to  one-half  the  arc  NB,  and  its  tangent  is  equal  to 


or 


-  --=  =  — ,  where  r  is  the  radius  of  the 


or 

SO       r 

circle.  From  this  relationship  it  is 
easy  to  locate,  mathematically,  the 
intersections  of  these  points  in  the 
projection.  This  is  of  value  in  de- 
termining the  points  where  the  lines 
extended  through  points  in  the 
southern  hemisphere  cut  the  pro- 
jection plane.  The  relative  posi- 
tions of  these  points  are  fixed;  the 
actual  distances  will  depend  upon 
the  scale  used. 

Since  all  the  intersections  be- 
tween meridians  and  any  parallel  lie 
at  the  same  distance  from  the 
center  (Fig.  10),  the  parallel  itself 
will  appear,  in  the  projection,  as  a 
circle  through  these  points,  conse- 
quently each  circle  in  the  figure 
represents  a  distance  10°  farther 
from  the  north  pole  than  the  adja- 
cent one.  The  points  A7  and  O 
(Fig.  9)  will  be  projected  at  M 
(Fig.  10),  consequently  the  line 
XOS  (Fig.  9)  will  be  projected 
upon  the  same  point.  Since  a  me- 
ridian is  the  intersection  of  a  sphere 
and  a  plane  passing  through  its 
north  and  south  poles,  and  since 
this  plane  must  contain  the  NOS 
line  which  is  vertical,  the  plane 
itself  must  be  vertical,  and  its  in- 
tersection with  the  sphere  must  be 
projected  as  a  straight  line  pass- 
ing through  the  center.  All  great 
circles,  therefore,  which  pass 
through  the  poles  of  the  sphere, 
appear,  in  stereographic  projection, 
as  straight  lines  passing  through 
the  center. 


N80° 


H' 

PIG.   10. 

FIGS.  9  AND  10. —  Method  of  locating 
points  in  stereographic  projection.  FIG.  9, 
Vertical  section  through  a  sphere;  FIG.  10. 
— Stereographic  projection  showing  positions 
of  parallels  and  meridians. 

N 


FIG. 


1 1 . — Tangent   relations 
sterecgraphic  projection. 


8 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  10 


PIG.   12. — Orthographic  projection  of 
a  circle  drawn  upon  A  sphere. 


10.  Circles  drawn  upon  a  Sphere  appear  as  Circles  in  Stereographic  Pro- 
jection.— One  of  the  chief  advantages  of  Stereographic  projection  over  other 

projections  is  the  fact  that  all  circles  traced 
upon  the  sphere  appear  as  true  circles  in 
the  drawing,  the  limiting  case  of  meridians 
appearing  as  straight  lines  being  the  case  of 
circles  with  centers  at  infinity. 

Let  Fig.  1 2  represent  a  sphere  in  ortho- 
graphic projection,  Fig.  13  a  section  .through 
'the  meridian  NP'E,  and  Fig.  14  a  stereo- 
graphic  projection  through  WE.  Upon  the 
sphere,  about  the  point  P'f  a  circle  is  de- 
scribed having  a  radius,  for  example,  meas- 
ured by  20°  of  its  surface.  The  upper  and 
lower  points  of  this  circle  will  appear,  in 
Fig.  13,  at  H  and  D,  and  the  center  atP'. 
The  triangle  HSD  (Fig.  13)  is  a  section  of 
>G  the  inclined  cone  HSD  of  Fig.  12.  It  has, 
by  construction,  a  circular  base  at  the  sur- 
face of  the  sphere,  and  has  its  apex  at  S.  If 
the  line  SD  be  extended  to  G  so  that  SG  = 
SH,  then  HSG  is  the  section  of  a  symme- 
trical cone  having  an  elliptical  base.  If, 
now,  this  cone  be  rotated  through  180°  on 
its  axis  SPf,  so  that  the  major  and  minor 
axes  are  parallel  to  their  former  positions, 
the  circle  HD  will  be  in  the  position  FG  (Fig. 
13).  All  sections  through  the  cone  parallel 
to  either  of  these  sections,  consequently,  will 
have  similar  circular  sections.  In  other 
words,  there  will  be  two  series 'of  circular 
sections  in  the  cone,  namely,  sections  paral- 
lel to  HD  and  to  FG.  The  latter  sections 
are  parallel,  also,  to  the  equator  WE;  for  if 
a  line  JD  be  drawn  parallel  to  FG,  we  have, 
by  construction,  the  angle  JDS  =FGS.  We 
also  have  FGS  =  DHS,  since  it  is  the  same 
angle  in  a  revolved  position.  The  angle 
JDS  lies  on  the  circumference  of  a  circle 
and,  by  geometry,  we  know  its  value  to  be 
one-half  the  arc  JS.  DHS  also  lies  on  the 

circumference,  and  its  value  is  one-half  the  arc  DS.     The  included  angles 
being  equal,  the  arcs  JS  and  DS  are  equal,  consequently  the  line  DJ  is 


FIG.   13. — Vertical  section  through 
JV  H  P'  D  E  S  of  preceding  figure. 


FIG.  14. — Stereographic  projection  of 
the  small  circle  H  D. 


ART.  11] 


STEREOGRAPHIC  PROJECTION 


at  right  angles  to  the  line  NS,  that  is,  it  is  horizontal,  and  any  geometrical 
figure  drawn  upon  the  plane  of  which  this  line  is  the  projection,  will  appear  as 
a  similar  figure  in  the  projection. 

From  this  demonstration  we  may  see  that  the  stereo  graphic  projection 
of  any  circle  which  may  be  described  upon  a  sphere  will  be  a  true  circle.1  The 
stereographic  projection  of  the  center  of  the  small  circle  (P,  Fig.  14)  will  not, 
however,  be  the  center  of  the  projected  circle  (c),  but  will  lie  somewhat  within 
it. 

The  explanation  here  given  will  apply  also  to  great  circles,  which  are 
likewise  projected  as  true  circles. 

ii.  Spherical  Angles  appear  in  Their  True  Values  in  Stereographic  Pro- 
jection.— Another  advantage  of  stereographic  projection  is  the  fact  that  the 


FIG.   15. — Perspective  view  of  sphere  and 
intersecting  planes. 


FIG.    16. — Geometric  relations  between  angles. 


angle  at  which  two  circles  cross  on  the  sphere  appears  in  its  true  value  in  the 
drawing. 

Let  P'fS  zndP'gj  (Fig.  15)  be  two  great  circles  on  the  sphere.  It  is  to  be 
proved  that  the  angle  fP'g,  which  lies  on  the  surface  of  the  sphere,  will  appear 
in  its  true  value  in  the  projection.  As  the  simplest  case  consider  first  one 
side  of  this  angle  to  be  formed  by  a  great  circle  passing  through  the  pole  AT; 
then  P'fg  is  a  right  angle.  Since  a  spherical  angle  is  measured  by  the  angle 
between  the  tangents  to  the  great  circles  which  form  the  angle,  AP'  and  BP', 
tangents  to  the  two  great  circles  P'fS  and  P'gj,  will  measure  the  angle  fP'g, 
whereby  fP'g  =  AP'B.  Now  the  stereographic  projection  of  P'  is  P,  and  since 
the  angle  AP'P  =  APPf,  as  may  be  seen  from  Fig.  16,  we  have  an  isosceles 

1  For  analytical  demonstration  see  Thos.  Craig:  A  treatise  on  projections.  U.  S.  Coast 
and  Geodetic  Survey,  Washington,  1882,  13-28,  187-191. 

For  graphical  demonstration  see  E.  Gelcich  und  F.  Sauter:  Kartenkunde  geschichtlich 
dargestellt.  Leipzig.  2te  Aufl.  von  Paul  Dinse,  1897,  42-44. 

See  also  V.  Goldschmidt:  Ueber  stereographische  Projection.  Zeitschr.  f.  Kryst.,  XXX 
(1899),  260-271. 


10 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  12 


triangle  in  which  AP' =  AP.  If  we  pass  a  plane  through  the  two  tangents, 
AP'  and  BP',  its  trace  on  the  horizontal  plane  will  be  the  line  AB,  which  must 
necessarily  lie  at  right  angles  to  the  line  AP,  for,  by  geometry,  the  trace  of  a 
tangent  plane  lies  at  right  angles  to  the  shortest  line  between  it  and  the 
center  of  the  sphere.  We  have,  now,  two  triangles,  AP'B 
and  APB,  in  which  one  side  (AB)  is  common  to  both, 
one  side  equal  in  each  (APf  =  AP),  and  one  angle  a  right 
angle  (P'AB=PAB  =  go°).  The  triangles,  consequently, 
are  equal;  PfB=PB,  and  the  angle  AP'B  =  APB.  The 
angle  in  the  projection,  therefore,  is  the  same  as  the 
angle  on  the  sphere. 

In  a  similar  manner,  another  right  triangle,  as  DP  A 
or  CPA,  Fig.  17,  may  be  proved  to  be  projected  in  its 
true  value.  The  algebraic  sum  of  APB  and  DP  A  or  CPA  (  =  DPB  or 
CPB),  being  thus  projected  in  its  true  value,  any  angle,  however  placed 
and  of  whatever  value,  will  also  so  appear. 

12.  Graphical  Solutions  of  Problems. — (i)  Given  a  pole,  to  find  the  corre- 
sponding great  circle.  Let  the  required  pole  be  30°  above  the  horizon  and  130° 
to  the  left  front.  Let  Fig.  18  be  a  vertical  section  through  the  sphere  along 


FIG.     1 8. — Vertical     section     through 
sphere,  showing  locations  of  points. 


FIG.   19. — Stereographic  projection  of 
preceding. 


the  130°  meridian,  then  P',  Fig.  18,  30°  above  F,  will  be  its  vertical  projection, 
and  P,  Fig.  19,  130°  to  the  front  and  on  the  parallel  through  P,  its  stereo- 
graphic  projection.  KT,  Fig.  18,  is  the  trace  of  the  plane  which  passes 
through  the  center  of  the  sphere  and  lies  at  right  angles  to  the  line  OP' . 
Its  intersection  with  the  surface  of  the  sphere  is  the  required  great  circle. 

In  the  vertical  section  (Fig.  18),  the  point  O  represents  the  line  GL  of  the 
Stereographic  projection — the  piercing  points  (G  and  L)  through  the  sphere, 
being  two  antipodal  points  on  the  great  circle.  The  point  /'  and  its  projec- 
tion 7,  90°  from  P,  represent  a  third  point  on  the  circle.  It  is  now  only 
necessary  to  pass  a  great  circle  through  the  three  points  G,  I,  L,  Fig.  19.  The 


ART.  12] 


STEREOGRAPH  1C  PROJECTION 


11 


center  of  the  circle  will  lie  half  way  between  7  and  the  projection  of  K' . 
The  latter  point,  however,  falls  too  far  to  the  left  to  make  it  possible  to 
determine  the  center  by  taking  half  the  distance,  IK.  If  one  uses  circles 
of  uniform  size  for  the  projection,  it  is  possible  to  construct  scales  giving 
the  positions  of  the  centers  for  various  great  circles,  a  method  used  by  Pen- 
field1  for  projection  circles  14  cm.  in  diameter.  The  center  may  be  located 
without  scales,  however,  since  it  must  lie  at  equal  distances  from  L,  I  and  G. 
With  these  three  points  as  centers,  describe  two  sets  of  equal  arcs,  such  as 

'),  Fig.  20.     If  the  arcs  drawn  with  7 


g',  I',  i',  and   g",  I",  i"  (or  g"',  I'", 


FIG.  20. — Method  for  locating  the  center  of  a  cir- 
cle when   three  points  upon  the  arc  are  given.    • 


FIG.  21. — Another  method  for  locating  the 
center  of  a  circle  when  three  points  upon  the 
arc  are  given. 


as  a  center  fall  on  opposite  sides  of  the  intersection  of  the  other  two  (if,  /' ', 
and  i",  I"),  connect  opposite  angles;  if  they  fall  on  the  same  side  (i",  I",  and 
*""'»  J'")>  connect  angles  on  the  same  side.  The  desired  center  (C)  is  where 
these  straight  lines  cross.  Another  method  of  finding  the  center  is  to  erect 
perpendiculars  to  two  chords  (GL  and  GI,  Fig.  21).  The  intersection  is  the 
desired  point. 

(2)  To  pass  a  great  circle  through  two  points  which  fall  within  the  equatorial 
circle.  Let  O  and  D,  Fig.  22,  be  any  two  points  within  the  equatorial  circle. 
The  center  of  the  projected  circle  passing  through  these  points  must  lie  at 
equal  distances  from  each,  consequently  it  must  lie  on  a  line  at  right  angles 
to  the  line  connecting  them.  Construct  this  line  (GC)  by  drawing  equal  arcs 
from  the  two  points,  and  connect  the  intersections.  Other  points  on  the  great 
circle  are  the  antipodal  points  to  O  and  D,  either  one  being  sufficient  to  de- 
termine it.  The  position  of  the  antipodal  point,  say  of  O,  Fig.  22,  can  be 
determined  by  making  use  of  an  auxilliary  great  circle.  Draw  a  vertical 
great  circle,  or  meridian,  through  O  (AOMB),  and  measure  the  elevation  of 

1  See  references,  page  16,  infra. 


12 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  12 


this  point  above  the  equator  by  making  use  of  the  projected  parallels.  In 
Fig.  22  this  distance  (A  to  O)  is  30°.  The  desired  antipodal  point  (E)  is 
30°  below  the  equator  on  the  same  meridian,  that  is,  180°  from  O.  Con- 


FIG.  22. — Construction  for  determining  the  cen- 
ter of  a  great  circle  passing  through  two  given 
points. 


FIG.  23. — Construction  for  determin- 
ing the  center  of  a  great  circle  through 
two  points,  one  of  which  lies  on  the 
equator. 


FIG.    24. — Construction   for   locating 
the  poles   of  a  given  great  circle. 


FIG.  25. — Projection  of  a  small  circle. 


struct  a  perpendicular  to  DE;  its  intersection  with  the  extension  of  GM  is  the 
desired  center  (C). 

(3 )  To  pass  a  great  circle  through  two  points ,  one  of  which  lies  on  the  equator. 
The  desired  circle  must  pass  through  O  and  D,  Fig.  23.     It  must  also  pass 


ART.  12] 


STEREOGRAPH  1C  PROJECTION 


13 


through  the  antipodal  point  of  D.  Since  the  latter  lies  upon  the  equator,  its 
antipodal  point  (D')  must  also  lie  upon  it.  Construct  the  vertical  great 
circle  or  meridian  DMD'  to  locate  D'.  The  -desired  center  must  lie  on  a  line 
at  right  angles  to  this  meridian,  that  is,  on  a  line  through  M,  intersecting 
the  equator  90°  from  D  and  from  ZX.  It  must  also  lie  on  the  medial  line 
between  O  and  D.  The  intersection  of  these  two  lines  marks  the  location  of 
the  center  C. 

(4)  To  find  the  poles  of  a  given  great  circle.     By  means  of  the  projected 
parallels,  measure  90°  each  way  from  the  point  where  the  great  circle  crosses 
the  bisecting  meridian.     Thus  the  distances  from  E  to  the  poles  P  andP'  (Fig. 
24)  are  each  90°. 

(5)  To  draw  a  small  circle,  its  size  and  the  location  of  its  center  on  the  sphere 
being  given.     Let  it  be  required  to  draw  a  small  circle  with  a  radius  of  30°,  and 


FIG.   26. — Vertical   section    of 
tical   small  circle. 


FIG.   27. — Construction  for  projecting  vertical 
small  circles. 


with  its  center  40°  above  the  equator  and  at  the  right  on  the  110°  meridian. 
Draw  first  the  110°  meridian  (AMb,  Fig.  25).  Since  the  center  of  the  desired 
circle  is  40°  above  the  equator  and  its  radius  is  30°,  its  lower  point  b  will  fall 
at  the  intersection  of  this  meridian  with  the  10°  parallel.  The  upper  point 
a  of  the  circle  will  fall  4O0+300  =  7O°  above  the  equator;  the  pole  P,  40° 
above  it.  The  actual  center  c  of  the  circle  in  the  projection  will  lie  half 
way  between  the  points  a  and  b. 

(6)  To  draw  a  vertical  small  circle  of  given  size.  Let  a'b',  Fig.  26,  be  a 
vertical  section  through  a  vertical  small  circle  of  60°  radius.  One  point 
(a,  Fig.  27)  can  be  located  60°  from  P  and  on  the  desired  meridian  PD  by 
means  of  the  parall  els.  Two  points,  /  and  g,  each  60°  from  P  and  on  the  equa- 
tor, represent  two  other  points  on  the  vertical  small  circle.  The  problem  now 
becomes  that  of  constructing  a  circle  through  three  points,  which  may  be  done 
as  in  Case  i.  If  the  vertical  small  circle  in  the  projection  is  an  arc  of  long 


14 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  13 


radius,  the  center  of  the  circle  fag  may  lie  off  the  paper.  In  such  cases  the 
line  may  be  drawn  best  by  means  of  the  curved  ruler  described  below  (Art.  13). 
(7)  To  measure  the  angles  of  a  spherical  triangle. — From  trigonometry  we 
know  that  a  spherical  triangle  is  one  formed  by  the  intercepts,  on  the  surface 
of  a  sphere,  of  a  triedral  angle  with  its  vertex  at  the  center.  As  the  angular 
distance  between  two  points  on  a  globe  is  measured  in  degrees  on  the  arc  of 
a  great  circle,  so  also  are  the  angles  in  the  projection  of  a  spherical  triangle 
measured  by  the  arcs  of  great  circles  at  a  distance  of  90°  from  the  angle. 

The  diedral  angles  of  the  triedral  angle 
are  the  angles  of  the  spherical  triangle,  and 
these  have  their  original  values  in  the  stereo- 
graphic  projections.  Thus,  to  measure  the 
angle  H'AD',  (Fig.  28),  draw  tangents  to  each 
circle  at  A,  and  measure  the  angle  between 
them  by  means  of  a  transparent  protractor. 
To  draw  an  accurate  tangent,  lay  off  equal 
distances  on  each  arc,  as  cc'  and  dd',  connect 
these  points,  and  draw,  through  A ,  lines  paral- 
lel to  the  chords  thus  located.  The  angle  dxcf, 
FIG.  28.-Measurement  of  the  pro-  between  the  chords,  is,  of  course,  the  same  as 

jection  of  a  spherical  triangle. 

the  angle  aAa'  between  the  tangents. 

A  much  simpler  method,  involving  the  use  of  a  stereographic  net,  is 
described  below. 

13.  Protractors  and  Scales. — As  mentioned  above,  the  process  of  making 
the  measurements  required  in  stereographic  projection  can  be  much  simplified 
by  the  use  of  suitable  protractors  and  scales.  So  long  ago  as  1867  there  was 
used,  in  the  U.  S.  Hydrographic  Office,  a  protractor  divided  into  degrees  by 
great  and  small  circles,  and  known  as  Professor  Chauvenet's  Great  Circle 


FIG.   29. — Curved  ruler,  after  Wulff  and  von  Fedorow.     2/7  natural  size.      (Fuess.) 

Protractor.  An  illustration  of  it  is  given  by  Sigsbee.1  Wulff,2  in  1893, 
used  the  stereographic  projection  in  showing  the  optical  properties  of  crystals, 
and  gave  an  illustration  of  a  curved  ruler  to  be  used  in  drawing  arcs  of  large 

1  Capt.  C.  D.  Sigsbee:  Graphical  methods  for  navigators.     U.  S.  Hydrographic  Office, 
Washington,  D.  C.,  1896.* 

2  Georg  Wulff:  Ueber  die  Vertauschung  der  Ebene  der  stereographischen  Projection  und 
deren  Anwendung.     Zeitschr.  f.  Kryst.,  XXI  (1893),  249-254.  . 


ART.  13] 


STEREOGRAPHIC  PROJECTION 


15 


circles.  Von  Fedorow,1  in  a  series  of  articles  on  determinative  methods 
beginning  the  same  year,  made  much  use  of  this  projection.  He  gave,  in 
his  first  paper,  a  mathematical  explanation  of  why  the  curve  in  the  curved 


FIG.  30. — The  von  Fedorow  net  for  stereographic  projection.      1/2  size  of  original. 


1  E.  von  Fedorow:  Universal-(Theodolith-}Methode  in  der  Mineralogie  und  Petro graphic. 
7.  Universalgcomctrischc  Untersuchungen.  Zeitschr.  f.  Kryst.,  XXI  (1893),  574-714. 

Idem:  Unwersal-(Theodolith-}Methode  in  der  .  Mineralogie  und  Petrographie.  II. 
Krystalloptische  Untersuchungen.  Ibidem,  XXII  (1894),  229-268. 

Idem:  Universalmethode   und   Feldspathstudien.     I.    Methodische    Verfahren.     Ibidem, 

XXVI  (1896),  225-262. 

Idem:  Universalmethode  und  Feldspathstudien.     II.     Feldspathbestimmungcn.     Ibidem, 

XXVII  (1897),  337-398. 

Idem:  Universalmethode  und  Feldspathstudien.  III.  Die  Feldspdthe  des  Bogoslovsk- 
schen  Bcrgreviers.  Ibidem,  XXIX  (1898),  604-658. 

Idem:  Umvcrsalgoniomcter  mil  mehr  als  zwei  Drehaxen  und  genaue  graphisclie  Redlining. 
Ibidem,  XXXII  (1899),  468-478. 

Idem:  Zur  Theorie  der  Krystattographischen  Proejctionen.  Ibidem.,  XXXIII  (1900), 
589-598. 


16 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  13 


ruler  is  always  the  arc  of  a  circle.  This  instrument  has  since  been  improved1 
and  now  possesses  a  scale  from  which  one  can  read  directly  the  curvature 
of  the  arc  (s,  Fig.  29). 

In  his  paper  in  1897,  von  Fedorow  published  a  stereographic  net  which 
greatly  simplifies  both  drawing  and  computation.  It  is  printed  in  pale  blue 
or  gray  ink  on  tracing  paper,  and  shows  divisions,  5°  apart,  of  two  sets  of 
stereographically  projected  great  circles  and  vertical  small  circles  at  right 
angles  to  each  other,  one  series  of  horizontal  small  circles,  and  one  of  vertical 
great  circles,  also  5°  apart.  The  original  net  is  20  cm.  in  diameter  and  is 
shown,  half  size,  in  Fig.  30.  It  is  used  by  placing  it  over  the  drawing  and 
pricking  through  to  locate  desired  points,  or  by  rotating  it  to  read  angles. 


The  graduation  on  the  Base  Line  gives  the  stereogiaphleally  projected  degrees. 
From  *  to  0  equals  the  chord  of  90 3 


40    *  5Q        60       70       80       90       8,0       70       60        $0        40          8 

Iliiilliiiliiiiiiiiiliiiiiiiiilii^ 


PIG.  31. — Protractor  used  by  Penfield. 


Further  directions  for  its  use  will  be  given  below.     A  similar  net  was  used  by 
Michel-Levy2  in  1894  and  later. 

In  1901  appeared  the  first  of  a  series  of  articles  by  Penfield3  on  stereo- 
graphic  projection;  a  series  which  has  done  more  than  any  other  publication 
in  English  to  bring  the  method  before  mineralogists  and  petrologists.  In 

1  E.  von  Fedorow:  Ueber  die   Anwendung  des  Dreispitzzirkels  fur  krystallographische 
Zwecke.     Zeitschr.  f.  Kryst.,  XXXVII  (1902-3),  138-142. 

2  A.  Michel-Levy:  Etude  sur  la  determination   des  fcldspaths  dans  les  plaques  minces. 
Paris,  1894. 

Idem:  Ibidem.     Deuxieme  fascicule,  1896. 
Idem:  Ibidem.     Troiseme  fascicule,  1904. 

3  Samuel  L.  Penfield:  The  sterso graphic  projection  and  its  possibilities  from  a  graphical 
standpoint.    Amer.  Jour.  Sci.,  XI  (1901),  1-24,  115-144. 

Idem:  On  the  use  of  the  stereographic  projection  for  geographical  maps  and  ^sailing  charts. 
Ibidem,  XQI  (1902),  245-276,  347-376. 

Idem:  On  the  solution  of  problems  in  crystallography  by  means  of  graphical  methods  based 
upon  spherical  and  plane  trigonometry.  Ibidem,  XIV  (1902),  249-284. 


ART.  13]  STEREOGRAPH  1C  PROJECT  I  OX  17 

his  first  paper,  Penfield1  gave  rather  an  elaborate  discussion  of  principles  and 
methods,  and  described  a  series  of  celluloid  and  paper  protractors.  His 
instruments  and  scales  include  (i)  a  protractor  (Fig.  31)  whose  circle,  14  cm. 
in  diameter,  is  divided  into  degrees,  and  whose  base  shows  the  stereographically 
projected  positions  of  these  points,  (2)  a  protractor  for  measuring  the  arcs  of 
great  circles,  and  consisting  of  a  series  of  vertical  small  circles  i°  apart,  (3)  a 


FIG.  32. — The  Wulff  stereographic  net.     1/2  size  of  original. 

protractor  giving  great  circles  and  vertical  small  circles  5°  apart,  and  (4)  a 
protractor  giving  great  circles  2°  apart.  Several  of  these  protractors  might 
well  be  combined  into  one,  as  is  done  is  the  Fedorow  net. 

Besides  these  protractors,  Penfield  made  a  series  of  very  useful  scales. 
One  gives  the  radii  of  stereographically  projected  arcs  of  great  circles,  and 
is  used  to  determine  the  centers  of  these  circles  in  the  projection  without  de- 
termining them  graphically.  Another  gives  the  radii  of  stereographically 
projected  arcs  of  vertical  small  circles,  and  a  third  gives  the  stereographic 
projections  of  the  intersections  of  a  vertical  great  circle  with  parallels,  up  to 

Samuel  L.  Penfield:  Op.  cil.,  XI  (1901),  138.  Penfield's  protractors  and  scales  are 
for  sale  by  E.  L.  Washburn  &  Co.,  New  Haven,  Conn. 


18  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  13 

and  including  156°  from  the  north.  The  latter  is  a  continuation  of  the  base 
line  of  Fig.  31.  A  curved  ruler,  with  the  curved  strip  made  of  wood,  thus  dif- 
fering from  the  von  Fedorow  model  mentioned  above,  is  also  described. 

In  1902,  Wulff,1  in  a  very  important  paper  on  the  optical  properties  of 
isomorphous  crystals,  published  the  lithograph  net  (Fig.  32)  which  is  most 
commonly  used  at  the  present  time.  Like  von  Fedorow's,  it  is  20  cm.  in 
diameter,  but  it  is  printed  on  heavy  paper,  and  over  it  is  laid  the  tracing 
paper  upon  which  the  drawing  is  to  be  made.  It  shows  the  stereographic 
projections  of  great  circles  and  of  vertical  small  circles,  2°  apart. 

Hutchinson,2  in  1908,  prepared  a  protractor  and  a  net  like  the  one  just 


\    1\0  \   3\0  \S\i  \ 

'1/0  '2/0   '.VO     ''4/0      ^     5/0          ^         ^o'                  ^'                     J2E                                              -^ 

\  8\0\8\0\4\0\ 

8/0     7/0     8/0       5/0           4/0                   3/-0                                          2^-"0 

/0%Vo%9/$l 

ZE 

OH 

-ft/- 

RO 

az                                                                                         5§- 

/  ///  /  /  1 

n~ 

1\\\\\\\\\     V       "\         ^               "^-^                          "~—-~.  ^ 

FIG.  33. — The  Hutchinson  stereographic  protractor. 

described  except  that  it  has  a  diameter  of  only  5  in.  (12.6  cm.).  For  the  use 
of  students,  and  where  extreme  accuracy  is  not  required,  the  5-in.  circle  is 
much  more  convenient  than  the  larger  one,  since  it  can  be  constructed  on  a 
sheet  of  paper  of  moderate  size  and  most  of  its  circles  can  be  drawn  with  an 
ordinary  pair  of  compasses.  The  protractor  (Fig.  33)  is  adapted  to  the  5- 
in.  circle.3  It  is  made  of  boxwood,  is  2.5  in.  in  width,  and  about  12  in.  in 
length.  The  intersection  of  the  edge  of  the  protractor  with  the  zero  line, 
which  extends  across  it  about  2  5/8  in.  from  one  end,  forms  the  center  of  the 
circle.  The  divisions  toward  the  shorter  end  represent  the  stereographic 
projections  of  every  second  degree.  The  longer  end  is  divided  into  degrees 
like  an  ordinary  rectangular  protractor  and  may  be  used  for  setting  off  angles. 
By  multiplying  these  divisions  by  two  it  likewise  gives  the  stereographic 
projections  of  points  lying  below  the  equator.  For  the  sake  of  clearness, 
the  finer  divisions  have  been  omitted  from  both  ends  in  the  figure. 

Johannsen,4  in  1911,  constructed  a  drawing-board  which  greatly  simpli- 

1  Georg    Wulff:    Untersuchungen  im  Gebiete  der  optischen  Eigenschaftcn    isomorpher 
Krystalle.     Zeitschr.  f.  Kryst.,   XXXVI  (1902),  1-28. 

2  A.  Hutchinson :  On  a  protractor  for  use  in  constructing  stereographic  and  gnomonic 
projections  of  the  sphere.     Mineralog.  Mag.,  XV  (1908),  93-112. 

3  In  a  letter  to  the  author,  Doctor  Hutchinson  states  that  protractors  may  now  be 
had  of  the  following  radii:  10  cm.  for  use  with  the  Wulff  net,  7  cm.  for  use  with  Pen- 
field's  circles,  5  cm.,  2  1/2  in.  as  stated  above,  and  i  1/2  in.  for  use  in  note-books.     The 
protractors,  graduated  on  boxwood,  are  manufactured  by  W.  H.  Harling,  47  Finsbury 
Pavement,  London. 

4  Albert  Johannsen:  A  drawing-board  with  revolving  disk  for  stereographic  projection. 
Jour.  Geol..  XIX  (1911),  752-755- 


ART.  14] 


STEREOGRAPH  1C  PROJECTION 


19 


fies  the  operation  of  rotating  a  stereographic  net.  In  it  is  combined,  on  a 
single  dial,  arcs  covering  all  vertical  and  horizontal  great  and  small  circles 
(Fig.  34).  Ordinarily,  in  stereographic  nets,  it  is  necessary  to  rotate  the  draw- 
ing above  the  net,  and  great  care  is  necessary  to  keep  the  two  accurately 
adjusted.  In  this  protractor  no  centering  is  necessary,  the  net  being  accu- 
rately centered  on  a  revolving  disk.  The  base,  which  is  a  drawing-board,  33 
by  43  cm.  in  size,  carries  a  net  20  cm.  in  diameter.  The  latter  is  composed 
of  two  semi-nets,  one  half  being  a 
Wulff  net,  the  other  half  drawn  to 
show  horizontal  and  vertical  small  cir- 
cles. The  figure  shows  divisions  only 
to  10°  although  both  halves  of  the  actual 
net  are  divided  to  2°.  This  drawing- 
board  is  inexpensive  and  is  adapted  to 
students'  use.  A  sheet  of  tracing 
paper  is  laid  above  the  net,  and  is 
fastened  to  the  board  by  means  of 
thumb-tacks. 

Points  are  located,  and  angles  and 
distances  are  measured  by  rotating 
the  disk,  curves  being  sketched  free- 
hand where  needed.  Drawings  made 
on  semi-opaque  tracing  paper  or 
cloth  will  readily  reproduce  by  photo- 
engraving. 

As  imilar  drawing-board,  constructed 
of  pasteboard,  was  described  later  by 
Noll.1 

14.  Calculating  the  Location  of  Points  in  Stereographic  Projection. — If 
it  is  intended  to  make  accurate  drawings  for  reproduction,  it  is  not  sufficient 
to  sketch  the  desired  curves  free-hand,  but  one  must  locate  the  centers  of  the 
circles  in  the  projection.  For  a  circle  of  small  radius,  the  point  may  be  deter- 
mined by  the  methods  shown  in  Figs.  20  and  21;  for  long  radii  a  curved 
ruler  may  be  used  or  the  distance  to  the  center  may  be  computed.  A  scale 
giving  the  computed  values  for  most  of  the  projected  circles  may  easily  be 
constructed  on  cardboard.  The  process  of  computing  the  points  is  simple 
since  a  stereographically  projected  degree  point,  such  as  b,  Fig.  35,  is  located 
at  a  distance  (Ob)  from  the  center  equal  to  the  radius  multiplied  by  the 
natural  tangent  of  half  the  angle  measured  on  the  arc  (ND) . 

For  example:  let  it  be  required  to  find  the  center  of  the  projection  of  a 
vertical  small  circle  having  a  radius  of  10°.  By  geometry,  the  angle  on  the 

1  F.  Noll.  Zeichenblock  fur  stereographische  Projektionen.  Centralbl.  f.  Min.,  etc., 
1902,  380-381. 


FIG.  34- — The  Johannsen   drawing-board   for 
stereographic  projection. 


20 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  14 


circumference  of  a  circle  includes  between  its  lines  an  arc  twice  as  great. 
In  Fig.  35,  the  point  b  is  the  stereographic  projection  of  the  point  D  on  the 
circumference,  the  latter  point  being  80°  from  N.  Let  this  point  represent 
the  upper  edge  of  the  desired  small  circle  on  the  sphere.  The  angle  OSb  — 

40°;  tan  OSb  =  ^;  Ob  =  OS  tan  OSb.     Ob  is  the  required  distance  to  the  inner 

line  of  the  circle  in  the  projection, 
OS  the  radius,  and  OSb  half  the 
angle  ND.  In  the  present  case  Ob 
=  tan  40° X  ioo  mm.  (the  circle  of 
reference  being  a  Wulff  net  20  cm. 
in  diameter)  =  0.8391X100  =  83.91 
mm.,  the  required  point  b.  Sim- 

_    GON 

ilarly 

£t 

X  ioo  =119.18  mm.,  the  required 
distance  to  the  outer  rim  of  the 
circle.  Upon  a  sphere  20  cm.  in 
diameter,  therefore,  the  diameter 
of  a  projected  circle  which  includes 
an  arc  of  20°  of  the  equator  is  aO  — #0=119.18  — 83.91  =35.27  mm. 

ab 
The  radius       =17.635  mm. 

I.    TABLE  GIVING  THE  CALCULATED  POSITIONS  OF  CENTERS  IN  THE  PROJECTION 

OF  VERTICAL  SMALL  CIRCLES,   MEASURED  IN   MM.  FROM  THE  POINT 

WHERE  IT  CROSSES  THE  MERIDIAN  WITHIN  THE  SPHERE 

Formula:  2x=  ioo  (tan  1/2  larger  arc  — tan  1/2  smaller  arc.) 


aSO=—  =  so°;aO=i.igiS 


FIG.  35. — Tangent  relations  in  stereographic 
projection. 


Radius  of 
circle 
on  sphere, 
degrees 

Radius  of 
projected 
circle, 
mm. 

Radius  of          Radius  of           Radius  of 
circle              projected               circle 
on  sphere,             circle,           ,  i  on  sphere, 
degrees                 mm.                  degrees 

Radius  of 
projected 
circle, 
mm. 

5 

8.850 

56 

148.260 

74 

348.745 

10 

17-635 

57 

I53--  985 

75 

373-205 

15 

26.792 

58 

160.035 

76 

401  .076 

20 

36-895 

59 

166.430 

77 

433  •  148 

25 

46.630 

60 

173.210 

78 

470.465 

30 

57-735 

61 

180.405 

79 

514.455 

35 

70.002 

62 

188.075 

80 

567.130 

36 

72.655 

63 

196.  260 

81 

63I-375 

38 

78.130 

64 

205.030 

82 

7II-538 

40 

83  .910 

65 

214.450 

83 

814.437 

42 

90  .  040 

66 

224.600 

84 

951-434 

44 

96.570 

67 

235-585 

85 

1143.007 

46 

Jo3-555 

68 

247.510 

86 

1430.069 

48 

in.  060 

69 

260.510 

87 

1908.115 

50 

iiQ-175 

70 

274.750 

88 

2863.627 

52 

127.995 

7i 

290.425 

89 

5728.999 

54 

137.640 

72 

307.770 

90 

CO 

55 

142.815 

73 

327.085 

ART.  14] 


STEREOGRAPHIC  PROJECTION 


21 


For  convenience  of  use  with  a  net  20  cm.  in  diameter,  the  radii  of  vertical 
small  circles  at  close  intervals  are  computed  above.  The  position  of  any 
center  is  found  by  laying  off  the  proper  distance  outward  from  the  point 
where  the  required  circle  cuts  the  meridian  through  its  center. 

II.    TABLE  OF  CENTERS  OF  GREAT  CIRCLES  MEASURED  FROM  THE  TRACE  OF  THE 

PROJECTED  CIRCLES 
Formula:  2X=  100  (tan  1/2  one  arc  +  tan  1/2  other  arc).    Fig.  36,  lettered  same  as  Fig.  35. 


j 

1 

Angle  between           Radius  of 
pole  (P)  and  N,          projected 

Angle          Radius  of 
between        projected 

Angle 
between 

Radius  of 
projected 

or  tilt  of  section              great 
from  equator,               circles, 

pole                great 
(P)  and  N,         circles, 

pole 

(P)  and  N, 

great 
circles, 

degrees                      mm. 

degrees              mm. 

degrees 

mm. 

5 

100.38 

56                178.83 

74 

362.79 

10 

101.54 

57                183.60 

75 

386.37 

15 

103-52 

58                188.70 

76 

4I3-35 

20 

106.41 

59 

194.16 

77 

444  -  54 

25 

110.34 

60 

200.00 

78 

480.97 

30 

115-47 

61 

206.  26 

79 

524.08 

-      35 

122.08 

62 

213.00 

80 

575-88 

36 

123.60 

63 

220.  27 

81 

639  •  74 

38                         126.90 

64 

228.12 

82 

718.53 

40                         I30-S4 

65 

236.62 

83 

820.55 

42 

134.56 

66 

245.86 

84 

956.67 

44 

139.02 

67 

255-93 

85 

H47-37 

46 

142.95 

68 

266.95 

86 

I433-56 

48 

149.20 

69 

279.04 

87 

1910.73 

50 

155-57 

70 

292.38 

88 

2865.37 

52 

162.42 

7i 

307-15 

89 

5729.87 

54 

170.13   . 

72 

323.61 

90 

00 

55 

174-34 

73                342.03 

The  following  table  gives  the  projected  positions,  as  measured  from  the 
center,  of  degree  points  in  the  southern  hemisphere.     Their  projections  lie 


FIG.  36. — Section  through  sphere,  showing  a  great  circle  (D  G)  tilted  30°  from  the  equator. 

beyond  the  limits  of  the  net.  Thus  Oa,  Fig.  36,  is  the  projected  position  of 
a  point  30°  south  of  the  equator.  Formula  for  a  circle  20  cm.  in  diameter; 
x=  100  times  tan  1/2  the  arc  on  circumference,  measured  from  TV. 


22 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  15 


III. 


TABLE   OF   DISTANCES   IN   MM.    FROM   THE    CENTER  OF  THE  SPHERE  TO  THE 
PROJECTED  POSITIONS  OF  POINTS  LYING  BELOW  THE  EQUATOR 


Point 
located 
y°  from  N, 
degrees 

Distance 
from 
center  of 
sphere, 
mm. 

Point 
located 
y°  from  N, 
degrees 

' 
Distance 
from 
center  of 
sphere, 
mm. 

Point 
located 
y°  from  N, 
degrees 

Distance 
from 
center  of 
sphere, 
mm. 

y  =  9o 

100.00 

150 

373-21 

166 

814.43 

95 

109.13 

151 

386.67 

167 

877.69 

100 

119.  18 

152 

401  .  08 

168 

951-44 

105 

130.32 

153 

416.53 

169 

1,038.54 

no 

142  .81 

154 

433-15 

170 

1,143.01 

115 

156.97 

155 

45I-°7 

171 

1,270.62 

120 

173.21 

156 

470.46 

172 

1,430.07 

125 

192  .  10 

157 

491.52 

173 

1,634.99 

I30 

214-45 

158 

514.46 

174 

1,908.  ii 

135 

241.42 

159 

539-55 

175 

2,290.38 

138 

260.51 

1  60 

567-13 

176 

2,863.63 

140 

274-75 

161 

597.58 

177 

3,818.85 

142 

290.42 

162 

631-38 

178 

5,729.00 

144 

307.77 

163 

669.  12 

179 

11,458.87 

146 

327.09 

164 

7H-54 

180 

00 

148 

348.74 

165 

759-58 

15.  Accuracy  of  Stereographic  Projection. — To  test  the  accuracy  of  meas- 
urements made  on  a  stereographically  projected  map,  as  compared  with 
distances  mathematically  computed,  Penfield1  made  a  number  of  determina- 
tions.    Using  a  circle  14  cm.  in  diameter,  he  found  that  in  spherical  triangles 
in  which  no  side  was  of  even  degrees,  the  angles  and  arcs  could  be  measured 
with  an  average  error  of  about  5  minutes,  the  maximum  error  in  twenty-one 
measurements  being  15  minutes.     On  a  map  of  the  hemispheres,  he  plotted 
the  locations  of  New  York  and  New  Orleans.     The  true  distance  between 
these  two  cities,  as  computed  from  their  latitudes  and  longitudes,  is  16° 
52'  or  116  1/2  statute  miles.     With  his  protractor  he  found  the  distance  to 
be  1 6°  53'  to  17°  8',  a  maximum  error  of  only  16  minutes  or  about  18  statute 
miles.     The  actual  size  of  the  map  of  the  United  States  in  the  drawing 
which  he  used  was  i  1/8  by  11/16  in.  (28X17  1/2  mm.),  and  the  distance 
between  the  two  cities  about  0.4  in.!     WulfP  gives  his  errors  of  reading 
as  averaging  ±  28'  on  a  circle  20  cm.  in  diameter. 

16.  Problems  solved  by  Means  of  a  Stereographic  Net. — The  use  of  the 

Johannsen  drawing-board,  or  any  Stereographic  net,  is  illustrated  best  by 
several  problems  which  are  given  to  show  the  simplicity  of  this  method,  as 
compared  with  the  solutions  previously  worked  out. 

(i)  Given  a  pole,  to  find  the  corresponding  great  circle.  Taking  the 
problem  given  in  Article  12,  Case  I,  we  have  a  pole  located  30°  above  the 
horizon  and  130°  to  the  left  front.  For  future  orientation,  upon  a  sheet  of 

1  Samuel  L.  Penfield:  Op.  tit.,  XI  (1901),  131. 

2  Georg  Wulff :  Op.  cit.,  17-18. 


ART.  16J 


STEREOGRAPH  1C  PROJECTION 


23 


tracing  paper,  fastened  to  the  drawing-board  (not  to  the  disk),  draw  vertical 
and  horizontal  lines  through  the  center.  Count,  on  the  circumference, 
130°  from  H  to  a  (Fig  37),  and  30°,  as  measured  by  horizontal  small  circles,  to 
p.  This  is  the  stereographic  projection  of  the  pole.  Count  90°  from  p  to 
b,  or  30°  from  M  to  b,  and  90°  from  a  to  d,  and  from  a  to  dr.  Three  points, 
d,  b,  and  d',  are  now  located  on  the  great  circle.  Rotate  the  dial  until  the 
separation  line  between  the  two  half  nets  (hereafter  called  the  HH'  line) 
falls  on  dMd'  and  the  b  point  lies  over  the  left  half  of  the  net  (hereafter 
called  the  /'  net).  Through  b  sketch  a  circle  following,  or  interpolated 


FIG.  37. — Stereographic  projections  of  problems  i  to  9. 

between,  curves  beneath.  If  a  very  accurate  line  is  desired,  use  a  curved 
ruler  (Fig.  29),  or  find  the  center  of  the  circle  from  Table  II,  Article  14,  or 
by  construction,  and  draw  with  a  compass. 

(2)  To  pass  a  great  circle  through  two  points  which  fall  within  the  equatorial 
line.    Let  e  and/,  Fig.  37,  be  these  points.     Rotate  the  disk  until  a  great 
circle  of  the  /'  net  passes  through,  or  is  equally  distant  from,  the  two  points. 
Sketch  the  curve. 

(3)  To  pass  a  great  circle  through  two  points,  one  of  which  is  on  the  equator. 
Let  g  and  h  (Fig.  37)  be  these  points.     Rotate  the  disk  until  the  HHf  line 
falls  on  //  and  the  J'  net  lies  beneath  g.     Sketch  the  great  circle  passing 
through  this  point. 


24 


MANUAL  OF  PETROGRAPIIIC  METHODS 


[ART.  16 


FIG.  38. — Method  for  rotating  the 
plane  of  projection. 


(4)  To  find  the  poles  of  a  given  great  circle.     Rotate  the  disk  until  a  great 
circle  of  the  J'  net  coincides  with  the  given  curve.     Count  90°  from  the 
curve  on  the  JJr  line  to  locate  the  pole  (curve  dbd'  and  point  p,  Fig.  37). 

(5)  To  draw  a  small  circle,  given  its  size  and  the  location  of  its  center  on 
the  sphere.     Let  the  center  (k)  be  located  in  the  same  position  as  in  problem 
5,  Article  12,  40°  above  the  equator  and  110°  to  the  right  front,  and  let  its 
radius  be  30°.     By  means  of  the  /  net  count  110°  from  H  to  i,  and  40°  i  to  k 
(Fig.  37).     This  is  the  projection  of  the  pole.     The  required  circle  cuts  the 
line  ik  at  a  distance  of  30  stereographically  projected  degrees  in  either  direc- 
tion, namely,  at  m  and  n.     The  center  of  the 
projected  circle  is  halfway  between  the  two 
at  c. 

(6)  To  draw  a  vertical  small  circle  of  given 
size.  Let  the  circle  be  one  of  35°  radius, 
with  its  center  on  a  meridian  30°  to  the 
right  of  H'.  Locate  its  center  at  q  (Fig.  37) 
and,  by  means  of  the  /  net,  locate  three  other 
points,  r  and  rf  on  the  equator  and  r"  on 
the  meridian,  each  35°  from  q.  Draw  rr"r' 
by  means  of  the  /'  net.  Owing  to  the  fact 
that  vertical  small  circles  are  cut  in  half  in 
the  Johannsen  drawing-board,  the  continua- 
tion of  the  arc  must  be  located  by  the  degree  marks  on  the  equator.  If  it 
is  desired  to  sketch  the  complete  circle,  the  disk  may  be  rotated  through 
1 80°. 

(7)  To  measure  a  spherical  triangle.    Let  the  triangle  be  std',   Fig.  37. 
On  the  /  net  count  90°  on  a  meridian  from  /  to  v,  and  90°  to  the  right  and 
left  of  w  (the  backward  extension  of  the  meridian)  to  y  and  h  on  the  equator. 
By  means  of  the  /'  net,  draw  a  great  circle  through  hvx.     The  part  of  this 
circle  (oo1)  cut  off  by  st  and  td'  may  be  measured  by  the  vertical   small 
circles  of  the  /'  net.     It  gives  the  value  of  the  angle. 

(8)  To  .measure  angular  distances  between  two  points.     Use  the  method 
given  in  the  latter  part  of  the  preceding  problem. 

(9)  To  change  the  plane  of  the  projection.1     For  drawing  maps  in  which 
a  particular  point  is  desired  in  the  center,  it  is  necessary  change  the  plane 
of  the  projection.    Let  it  be  desired  to  move  the  point  P  (Fig.  38),  which 
lies  1 60°  to  the  right  front  and  20°  above  the  equator,  to  the  center  of  the 
projection  (P).     Rotate  the  net  until  the  //'  line,  which  is  the  trace  of  a 
vertical  plane,  passes  through  P.     In  this  position  the  line  HH'  represents 
the  axis  about  which  the  sphere  must  be  rotated  to  bring  the  point  P  to  the 
center.     During  the  rotation  every  other  point  upon  the  sphere,  such  as 

1  Georg  Wulff :  Ueber  die  Vertauschung   der  Ebene  der  stereo graphischen  Projection  und 
deren  Anwendung.     Zeitschr.  f.  Kryst.,  XXI  (1892-3),   249-254. 


ART.  17] 


STEREOGRAPHIC  PROJECTION 


25 


a  or  b,  will  be  turned  through  the  same  angle  and  about  the  same  axis,  con- 
sequently in  vertical  planes  at  right  angles  to  it.  Since  vertical  planes 
appear  in  stereographic  projection  as  vertical  small  circles,  it  is  only  necessary , 
for  the  rotation  of  any  point,  to  count  along  its  vertical  small  circle  the  same 
number  of  degrees,  as  from  P  to  P' ',  for  example  a-a',  b-bf,  etc.  Should  a 
point  lie  such  a  short  distance  above  the  equator  that  the  rotation  will 
bring  it  below,  its  projection  will  appear  beyond  the  periphery  of  the  net, 
as  d  at  d'. 

17.  Various  Accessories  used  in  Stereographic  Projection. — When  making 
drawings  upon  thick  paper,  it  is  not  always  convenient  to  transfer  points  by  pricking 
through  a  von  Fedorow  tracing-paper  net.  For  such  transfers,  a  three-point 


PIG.  39. — Von  Fedorow's  three-point  compass.     1/2  natural  size.     (Fuess.) 

compass1  (Fig.  39)  is  useful.  Two  of  the  points  are  set  to  marks  appearing  on 
both  drawing  and  net,  and  the  third  to  the  one  which  is  to  be  transferred.  A 
more  rigid  compass,  designed  by  Hutchinson,2  is  shown  in  Fig.  40. 


A*  B 

FIG.  40. — Hutchinson's  three-point  compass. 

A  simple  protractor,  valuable  as  an  accessory  to  the  drawing-board  described 
above,  may  be  constructed  by  students  for  their  own  use  by  drawing  upon  a  sheet 
of  cardboard  the  stereographically  projected  degrees  beyond  the  equatorial  circle 
of  20  cm.  The  values  given  in  Table  III,  Article  14  may  be  used.  On  a  scale 
30  in.  in  length,  the  degrees  up  to  165  can  be  plotted.  Beyond  that,  the  distances 
rapidly  increase. 

For  the  accurate  plotting  of  spherical  triangles,  a  Nikitin3  hemisphere  (Fig.  41), 
with  movable  graduated  circles,  is  valuable,  although  for  teaching  purposes,  a 

1  E.  von   Fedorow:  Ueber  die  Anwendung  des   Dreispitzzirkels  fiir  krystallographische 
Zwecke.    Zeitschr.  f.  Kryst.,  XXXVII  (1902-3),  138-142. 

2  A.  Hutchinson:  On  a  protractor  for  use  in  constructing  stereographic  and  gnomonic 
projections  of  the  sphere.     Mineralog.  Mag.,  XV  (1908),  93-112. 

3  W.  Nikitin :  Halbspharoid  zur  graphischen  Lb'sung  bei  A  nwendung  der  Universalmethode. 
Zeitschr.  f.  Kryst.,  XL VII  (1910),  379-381. 


20 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  17 


wooden  sphere,  12  to  16  in.  in  diameter  and  covered  with  blackboard  paint,  will 
answer.  In  addition,  parallels  and  meridians,  spaced  10°  apart,  should  be  shown 
by  very  narrow,  white  lines  and  the  intermediate  degrees  by  dots  on  the  equator. 
A  narrow,  graduated,  brass  strip,  attached  by  single  screws  at  the  north  and  south 
poles  and  bent  to  follow  the  contour  of  the  globe,  is  an  additional  help. 

Another  convenient  class-room  accessory  is  a  Wulfing1  wall-chart  for  stereo- 
graphic  projection.  It  consists,  in  its  latest  form,  of  a  plate  of  ground-glass  over- 
lying a  Wulff  net  70  cm.  in  diameter.  The  net  is  mounted  on  pasteboard  and  pro- 


R< 


FIG.  41. — Nikitin's   porcelain  hemisphere  for  graphical  representation  of  properties  of  crystals. 

1/4  natural  size.     (Puess.) 

jects  beyond  the  edges  of  the  frame  so  that  it  may  be  rotated  conveniently.  The 
ground-glass  is  hinged  at  some  distance  below  the  net  whereby,  when  tilted  forward, 
crayon  marks  upon  it  will  show  clearly,  but  the  net  will  not  be  seen  (Fig.  42). 

Figs.  43  and  44  represent  models  of  another  apparatus  constructed  by  Professor 
Wiilnng2  and  useful  in  showing  first,  the  relation  between  a  crystal  and  the  projec- 
tion sphere  (Fig.  43),  and  second,  the  projection  upon  the  plane  of  the  points  thus 
located  on  the  sphere  (Fig.  44).  The  method  of  using  the  model  is  clear  from  the 
illustrations. 

1  E.  A.  Wiilfing:  Wandtafeln  fur  stereographische  Projektion.  Centralbl.  f.    Min.,  etc., 
1911,  273-275. 

2  Idem:  Modell  zur   Erlauterung  der  stereographischen  Projektion.     Centralbl.  f.  Min., 
etc.,  1911,  749-752. 


ART.  17] 


STEREOGRAPHIC  PROJECTION 


27 


PROBLEMS 

Let  the  equatorial  plane  be  the  plane  of  the  projection. 

(a)  By  means  of  a  Wulff  net  or  a  Johannsen  drawing-board,  pass  a  great  circle 
through  two  points,  A  and  B,  one  lying  on  the  7oth  meridian  (left  side)  and  30° 
above  the  equator  (outer  circle  of  net),  the  other  on  the  i2oth  meridian  (left 
side)  and  60°  ab°ve  the  equator. 

(b)  What  is  the  angular  distance  between  these  two  points? 

(c)  Measure  the  angle  which  the  plane  passing  through  these  points  makes 
with  the  equatorial  plane. 


FIG.  42.  FIG.  43. 

FIG.  42. — The  Wulfing  wall  chart  for  stereographic  projection.     (Krantz.) 
FIGS.  43  AND  44. — Wulfing  projection  model.     (Krantz.) 


FIG.  44- 


(d)  Measure  the  angular  distance  between  this  plane  and  the  axis  forming  the 
center  of  the  projection. 

(e)  What  is  the  angular  distance  between  the  first  point  and  the  center  of  the 
projection? 

( f)  Draw  another  plane  passing  through  the  point  A  and  through  a  point  on 
the  60°  meridian  (right  side)  and  40°  above  the  equator. 

(g)  What  angle  does  this  plane  make  with  the  equatorial  plane? 

(h)  What  angle  does  the  great  circle  formed  by  the  second  plane  make  with  the 
great  circle  of  the  first  plane? 

(i)    Locate  the  poles  of  each  of  the  two  planes. 

( j)  What  is  the  angle  between  the  poles  (that  is,  the  angle  between  the  two 
planes)  ? 


28  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  17 

REFERENCES 

Besides  the  articles  mentioned  in  the  preceding  chapter,  the  books  and  papers  below 
may  be  of  assistance. 
1866.  A.  v.  Lang:  Lehrbuch  der  Krystallo graphic,  Wien,  1866.* 

1871.  E.  Reusch:  Bezeichnung  der  Hemiedrie  bei  Anwendung  der  stereo graphischen  Projec- 

tion.    Pogg.  Ann.,  CXLII  (1871),  46-54. 

1872.  Idem:  Zur  Lehre  von  den  Krystallz-willingen.     Pogg.  Ann.,  CXLVII  (1872),  569- 

589. 
1 88 1.  Idem:  Die  stereographische  Projektion.    Leipzig,  1881.* 

1873.  F.  A.  Quenstedt:  Grundriss  der  Krystallographie.     Tubingen,  1873.* 
1881.  Th.  Liebisch:  Geometrische  Krystallographie,  Leipzig,  1881,  116-134. 
1884.  A.  Brezina:  Methodik  der  Krystallbestimmung.     Wien,  1884.* 

1887.  V.  Goldschmidt:  Ueber  Projection  und  graphische  Krystallber echnung.     Berlin,  1887.* 
1893.  B.  Hecht:  Anleitung  zur  Krystallberechnung.    Leipzig,  1893.* 

1897.  E.  Gelcich  und  F.  Sauter:  Kartenkunde  geschichtlich  dargestellt.    Leipzig,  2  Aufl., 
1897,  38-54. 

1904.  Rosenbusch  und  Wiilfing:  Mikroskopische  Physiographic.     Stuttgart,  4  Aufl.,  1904, 

I-i. 

1905.  P.  Groth:  Physikalische  Krystallographie,  Leipzig,  1905,  314. 

1 91 1.  H.  E.  Boeke:  Die  Anwendung  der  stereographischen  Projektion  bei  kristallographischen 
Untersuchungtn.     Berlin,  1911. 


CHAPTER   III 
A  FEW  PRINCIPLES  OF  OPTICS 

18.  The  Nature  of  Light. — Before  describing  the  petrographic  microscope 
and  the  methods  of  its  use,  it  will  be  necessary  to  discuss  briefly  a  few 
elementary  principles  of  optics.1     Using  the  language  of  the  elastic   solid 
theory  for  descriptive  purposes,  without  implying  it  to  be  an  accepted 
theory  of  the  actual  physical  facts,  we  may  say  that  light  consists  of  vibra- 
tions of  some  kind  in  an. all-pervading  medium  which  we  call  ether.     What 
the  exact  nature  of  these  vibrations  is  we  do  not  know,  although  we  do 
know  that  they  follow  the  laws  of  wave-motion.     It  is  quite  probable  that 
there  actually  is  a  rapid  periodic  change  in  the  magnetic  and  electric  condi- 
tion of  the  ether.     This  electromagnetic  theory,  as  it  is  called,  is  the  most 
recent  one,  and  in  it,  like  in  the  former  generally  accepted  undulatory  theory 
of  Huygens,  the  periodic  oscillations  take  place  at  right  angles  to  the  direc- 
tion of  transmission  of  the  ray. 

19.  Corpuscular  or  Emission  Theory. — It  was  supposed  by  Newton 
that  light  consisted  of  innumerable  small  particles  sent  out  with  extreme 
rapidity  by  all  luminous  bodies.     He  thought  that  these  small  particles 
could  pass  freely  through  all  transparent  bodies  and  into  the  eye,  where 
they  produced  the  sensation  of  light  by  their  impact  upon  the  optic  nerve. 
Bennett2  reasoned  that  if  such  great  numbers  of  particles,  even  though  ex- 
tremely minute,  were  actually  sent  out  by  a  luminous  body,  they  should 
have  some  effect  of  deflection  upon  a  suspended  body,  yet  he  found  that 
when  light  was  concentrated  by  mirrors  and  lenses  and  was  directed  against 
a  most  delicate  balance  made  of  a  fragment  of  straw  suspended  horizontally 
from  a  single  spider  web,  not  the  slightest  motion  due  to  the  impact  of  the 
light  particles  appeared.3 

The  shooting  forth  of  light  particles,  under  the  corpuscular  theory,  was 
compared  with  the  movement  of  a  projectile,  and  refraction  was  supposed 
to  be  due  to  forces  of  attraction  or  repulsion  in  the  medium  into  which  the 
particles  passed.  If  the  medium  were  denser  than  the  one  from  which  the 
light  came,  the  rays  were  supposed  to  be  more  attracted,  consequently 

1  See  General  Bibliography  at  the  end  of  the  chapter. 

2  Rev.  A.  Bennett:  A  new  suspension  of  the  magnetic  needle,  intended  for  the  discovery 
of  minute  quantities  of  magnetic  attraction,  etc.     Phil.  Trans.  Roy.  Soc.,   London,  Pt.  I, 
1792,  81-98. 

3  In  this  connection  see  the  modern  measurements  of  light  pressure  by  Lebedew  and  by 
Xichols  and  Hull  which  show  that  light  does  actually  exert  a  pressure. 

29 


30  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  20 

an  oblique  ray  would  be  bent  toward  the  normal.  At  the  same  time  this 
greater  attraction  should  have  the  effect  of  augmenting  the  velocity  of  the 
particles.  It  was  shown  experimentally  by  Foucault  that  this  is  not  the 
case,  but  that  the  velocity  of  light  in  water  is  less  than  it  is  in  air.  It  has 
been  shown  definitely  that  the  velocity  of  light  decreases  as  the  index  of 
refraction  increases. 

*2o.  The  Undulatory  or  Wave  Theory  of  Huygens. — The  Dutch  astrono- 
mer and  physicist  Huygens1  was  the  first  to  oppose  the  emission  theory, 
although  it  was  supported  by  such  men  as  Laplace,  Biot,  and  Brewster. 
He  suggested  that  light  is  due  to  wave-motion,  a  theory  which  fell  into 
disuse  but  was  revived  after  nearly  a  century  and  gradually  gained  ground, 
especially  through  the  work  of  Thomas  Young2  and  Augustin  Fresnel. 
Although  Huygens  had  stated  that  light  is  due  to  a  vibratory  motion  in 
the  ether,  yet  on  this  theory  he  was  unable  to  account  for  the  phenomenon 
of  polarization  which  he  had  discovered.  Thirteen  years  before,  Hooke3 
had  defined  light  as  due  to  quick  and  extremely  short "  vibratile"  movements, 
and  later4  suggested  that  they  might  be  transverse,  although  he  did  not 
prove  it.  It  was  not  until  Young5  suggested  and  Fresnel6  demonstrated 
that  the  vibrations  actually  take  place  in  a  direction  transverse  to  the 
direction  of  transmission  that  the  theory  gained  ground.  According  to 
the  elastic-solid  theory,  the  speed  of  propagation  depends  upon  the  elasticity 
and  density  of  the  medium  through  which  it  is  passing,  consequently  the 
greater  the  elasticity  and  the  less  th.e  density,  the  greater  the  velocity. 

A  more  recent  theory,  although  also  dependent  upon  undulatory  or 
wave-motion,  is  the  electromagnetic  theory. 

21.  The  Electromagnetic  Theory. — Recent  researches  in  regard  to 
electromagnetic  waves  seem  to  show,  without  doubt,  that  light  is  due  to 
waves  of  the  same  character.  This  theory,  fundamentally  purely  electrical, 

1  Christian  Huygens:  Op.  cit.  in  General  Bibliography  at  end  of  chapter. 

2  Thomas  Young:  On  the  theory  of  light  and  colours.     Bakerian  lecture,  read  Nov.  12, 
1801.     Phil.  Trans.  Roy.  Soc.,  London,  XCII  (1802),  12-48. 

Idem:  An  account  of  some  cases  of  the  production  of  colours,  not  hitherto  described. 
Read  July  i,  1802.  Ibidem,  XCII  (1802),  387-397. 

Idem:  Experiments  and  calculations  relative  to  physical  optics.  Bakerian  lecture,  Nov. 
24,  1803.  Ibidem,  XCIV  (1804),  1-16. 

The  above  three  papers  reprinted  by  Henry  Crew  in  The  wave  theory  of  light,  Memoirs 
by  Huygens,  Young  and  Fresnel.  New  York,  (1900),  47-76. 

3  Hooke:  Micrographia,  1665,  15.* 

4  Idem:  Lecture  on  Light* 

5  Thomas  Young :  Jan.  12,  1817.* 

6  Augustin  Fresnel:  Memoir e  sur  la  double  refraction.     Mem.  Acad.  France,  VII  (1827) 
45-176. 

Idem:  Ueber  die  doppelte  Strahlenbrechung.  (Translation  of  preceding.)  Pogg.  Ann., 
(1831),  372-434;  494-560. 


ART.  23]  A  FEW  PRINCIPLES  OF  OPTICS  31 

was  proposed  by  Maxwell,1  who  supposed  that  there  is  an  intimate  con- 
nection between  the  vibrations  constituting  light  and  electricity.  He  said: 
"The  agreement  of  the  results  seems  to  show  that  light  and  magnetism  are 
affections  of  the  same  substance,  and  that  light  is  an  electromagnetic  dis- 
turbance propagated  through  the  field  according  to  electromagnetic  laws." 
Reflection  and  refraction  of  electromagnetic  waves  were  first  discussed  by 
Lorentz,2  and  later  by  J.  J.  Thomson,3  Fitzgerald,4  Glazebrook,5  ^nd 
Lord  Rayleigh.6 

So  far  as  we  are  concerned,  in  the  explanation  of  the  phenomena  of  light, 
it  will  be  sufficient  to  regard  it  simply  as  wave-motion  which  transmits  energy, 
but  not  matter,  by  means  of  oscillations  taking  place  in  the  ether  at  right 
angles  to  the  direction  of  propagation. 

22.  The  Ether. — What  the  ether  actually  is,  or  what  its  properties  are, 
we  do  not  know.     It  is  generally  assumed   to  be  a  medium  which  occurs 
everywhere,  filling  intermolecular  space  as  well  as  extending  through  inter- 
stellar regions. 

WAVE- MOTION 

23.  The   Movements   of   Oscillation. — Assuming   that   light   is    trans- 
mitted by  wave-motion,  it  will  be  well  to  consider  next  what  wave-motion  is, 
and  how  the  ether  is  affected  by  different  waves  and  different  combinations 
of  waves. 

If  a  particle  moves  in  a  certain  direction  from  a  point  of  equilibrium,  it 
will  move  with  gradually  diminishing  velocity  until  it  reaches  its  position 
of  maximum  swing.  It  will  pause  there  a  moment,  and  then  will  return  with 
gradually  increasing  velocity  to  its  position  of  rest.  Since  it  is  moving  with- 
out friction,  it  will  pass  beyond  this  point  of  rest  with  gradually  decreasing 
velocity  until  it  has  reached  a  point,  in  the  opposite  direction,  equal  to  its 
first  swing.  Here  it  will  pause,  will  then  return  with  increasing  velocity,  and 
so  on.  The  retardation  and  acceleration  of  the  motion  is  such  as  would  be 
seen  by  viewing,  in  the  plane  of  its  rotation,  a  particle  moving  uniformly 
around  a  circle.  Thus  if  the  particle  a,  Fig.  45,  moves  uniformly  around 

1  J.  Clerk  Maxwell:  A   dynamical  theory  of  the  electromagnetic  field.     Phil.  Trans.  Roy. 
Soc.,  London,  CLV  (1865),  459.* 

2  H.  A.  Lorentz :  Ueber  die  Theorie  der  Reflexion  und  Refraction  des  Lichtes.     Schlomilch's 
Zeitschr.,  XXII  (1877),  1-30,  205-219. 

3  J.  J.  Thomson:  On  Maxwell's  theory  of  light.     Phil.  Mag.,  IX  (1880),  284-291. 

4  G.  F.  Fitzgerald:  On  the  electromagnetic  theory  of  the  reflection  and  refraction  of  light. 
Phil.  Trans.  Roy.  Soc.,  London  for  1880,  CLXXI  ( 1 88 1),  691-711. 

5  R.  T.  Glazebrook:  On  some  equations  connected  with  the  electromagnetic  theory  of  light. 
Read  1881.     Proc.  Cambridge  Phil.  Soc.,  IV.  (1883),  155-167. 

6  Lord  Rayleigh:  On  the  electromagnetic  theory  of  light.     Phil.  Mag.,  XII  (1881),  81- 


32 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  23 


the  circle,  it  will  take  successively  the  positions  b,  c,  d,  e,  f,  .  .  .  i  .  .  k  .  .  .  a, 
equally  distant  from  each  other,  and  if  it  is  viewed  from  a  point  in  the  plane 
of  the  paper,  the  particle  will  appear  to  vibrate' along  the  diameter  of  the 
circle.  After  reaching  the  point  e,  the  motion  will  appear  backward  on  the 
diameter.  Since  the  movement  forward  and  backward  occurs  at  regular 
intervals  of  time,  it  is  said  to  be  periodic.  The  circle  abc  .  .  k.p  is  called  the 
circle  of  reference,  or  the  auxiliary  circle. 

The  equation  of  displacement  in  a  circle  is  as  follows : 

Let  a  =  aOd,  sin  <*=-„-;  (Fig.  45). 
Od 

But  Od  =  radius  =  r,  therefore  d'd  (the  displacement)  =  r  sin  a.     Also  let 
/  =  the  time  required  for  the  particle  to  move  one  division  on  the  circle. 


FIG.  45. —  Movement  of  a  particle 
around  a  circle. 


FIG.  46. — Velocity  of  a  particle  around 
a  circle. 


co  =  the  angle  through  which  this  particle  moves  in  a  unit  of  time.     Then  since 
sin  a  =        ,  d'd  =  r  sin  a,  and  the  arc  ad  =  ut  =  the  value  of  the  angle  a,  we  have: 

the  amplitude  (Oa  =  r)   times  the  sine  of  co/,  or: 
d  =  r  sin  cot  (i) 


The  displacement  (d'd) 


When  co/  =     o°  or  180°,  the  sin  co/  =  o  and  the  displacement  =  o. 

When  «/  =  90°  or  270°,  the  sin  co/  =  i  and  the  displacement  =  a. 

The  equation  of  the  velocity  of  any  point  in  a  circle  is  derived  as  follows: 

Let  the  particle  be  at  c  (Fig.  46)  on  the  auxiliary  circle,  and  let  cd,  tangent  at 
c,  represent  the  velocity.  Resolve  cd  into  two  components  cf  and  ce,  parallel  and 
at  right  angles  to  Ok.  The  two  right-angled  triangles  c'cO  and  ecd  are  similar,  for 
c'cO  +  Oce  =  90°  and  ecd  +  Oce  =  go0,  therefore  c'cO  =  ecd  and  therefore,  also, 

ed 
the  other  angles  are  equal,  and  edc  =  a,  cos  a  =     „  and  ed  =  cd  cos  a. 

Substituting  v  =  velocity  at  c,  and  vf  =  velocity  projected  on  Ok,  we  have 

Vr   —  V  COS  a  (2) 

Since  v  is  a  constant,  the  velocity  at  any  point  on  the  diameter  Ok  is  propor- 
tional to  the  cosine  of  the  corresponding  arc.  Also  (Fig.  48)  the  velocity  at  any 


ART.  25]  A  FEW  PRINCIPLES  OF  OPTICS  33 

point,  as  B',C',D',  etc.,  is  proportional  to  the  corresponding  distance  A  'B'  ,  A'C' 
A'D',  etc. 

When  «  =  o°  or  180°,  v'  =  v.     It  is  the  maximum  value. 

When  «  =  90°  or  270°,  the  velocity  equals  zero. 

Comparing  (i)  and  (2)  we  see  that  when  the  displacement  (i)  is  at  its  maxi- 
mum, the  speed  (2)  is  zero,  and  vice  versa. 

24.  Simple  Harmonic  Motion.  —  Simple  harmonic  motion  is  the  name 
given  to  such  motion  as  that  which  apparently  takes  place  backward  and 
forward  along  the  diameter  of  a  circle  (me,  Fig.  45)  when  looking  in  the  plane 
of  the  circle  at  a  steady  motion  around  its  periphery.  Such  motion  is  periodic, 
for  it  repeats  itself  at  regular  intervals.  The  distance  from  the  position  of 
rest  of  the  particle  to  the  limit  of  its  movement  is  called  the  amplitude  (Oe). 
The  period  is  the  interval  of  time  which  elapses  between  two  successive 
passages  of  a  particle  through  a  certain  point  in  a  certain  direction.  In  Fig. 
45  the  period  is  O  to  e  to  m  to  O.  The  phase  is  the  fraction  of  a  period  which 
has  elapsed  since  the  particle  last  passed  through  the  position  of  rest.  When 
it  is  farthest  from  O  on  the  positive  side,  it  is  said  to  be  in  its  position  of 
maximum  positive  elongation;  when  farthest  from  0  on  the  negative  side,  of 
negative  elongation. 


25.  Isochronism  and  Angular  Velocity.  —  When  a  particle  moves  in  a 
circular  path,  its  velocity  of  rotation  may  be  measured  by  the  distance 
traveled  divided  by  the  time,  or  it  may  be  measured  by  the  angle  through 
which  a  particle  at  unit  distance  passes  in  a  unit  of  time.  The  latter  measure- 
ment is  called  the  angular  velocity  and  is  indicated  by  aj. 

Let  T   =  the  time  of  a  complete  oscillation. 

2  TT  =  the  circumference  of  a  circle  having  a  radius  of  unity. 

Then  o,  =  y.  (3) 

This  is  the  equation  for  the  angular  velocity. 

If  a  is  constant,  T  also  must  be  constant.  That  is,  in  simple  harmonic  motion, 
the  period  is  independent  of  the  amplitude.  In  other  words,  the  particle  will  per- 
form its  oscillations  in  equal  periods  of  time  irrespective  of  its  amplitude.  It 
will  vibrate  isochronously. 

The  angular  velocity  may  be  expressed  in  another  way.  The  velocity  of  any 
other  particle  on  the  same  radius,  but  at  a  distance  of  r't  will  be  r'a>.  That  is, 

v  =  r'<a,  «  =  —  7'     Expressed  in  words  —  the  angular  velocity  of  a  particle  is  its  ve- 

locity divided  by  its  distance  from  the  center. 

We  have  further;  at  the  end  of  any  period  of  time  /  the  particle  c  will  have  moved 
through  the  angle  a  (Fig.  47),  therefore 

27T/ 
a=-  (4) 


34 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  26 


Draw  sc  and  cc'  parallel  to  Oa  and  Ok,  whereby 


and 


cc       d 

sin  a  =  —  =  -- 
r       r 


d  —  r  sin  a, 


(5) 
(6) 


in  which  d  is  the  distance  through  which  the  particle  has  moved  from  the  position  of 
rest. 

Substitute  the  value  of  a  from  (4)  in  (6), 


=  r  sn 


(7) 


FIG.  47. — Angular  velocity. 


This  equation  gives  the  position  of  any  point, 
whose  displacement  is  d,  in  terms  of  the  ampli- 
tude (r),  the  time  of  one  period  (T),  and  the 
time  since  the  beginning  of  the  movement  (/). 

If  a  certain  interval  of  time  (ti)  had  already 
elapsed  before  the  particle  was  set  in  motion, 
the  remaining  time  equals  t-ti,  and  the  equation 
becomes 


(8) 


(9) 


=  r  sm     — 


We  also  have 

cos  a  —  — ,  Oc'  =  r  cos 


«  =  r  cos  |^y  /J. 


But  the  cosine  of  an  angle  is  equal  to  the  sine  of  90°  plus  the  angle,  therefore 

j,    tj—r    0111    T     ^    ^    ~T   ")•  (lo) 


Oc'  =  r  sin  (-  +  -~r /)=r  sin-=r  (—  -f- 
2         J.  J.       4 


Equations  (8)  and  (9)  are  the  equations  for  the  lateral  displacement  of  the 
particle. 

26.  Harmonic  Curves. — A  particle  may  be  subjected  to  two  or  more 
movements  at  the  same  time.  If  a  particle,  moving  in  simple  harmonic 
motion  along  a  line,  be  also  subjected  to  a  uniform  motion  of  translation  in  a 
direction  perpendicular  to  that  line,  the  resultant  curve  is  called  the  harmonic 
curve. 

Let  G  and  S,  in  the  circle  of  reference  (Fig.  48),  be  the  positions  of  maxi- 
mum elongation  of  a  particle  moving  along  the  diameter.  Let  the  distances 
AB,  BC,  CD,  etc.,  on  the  circle,  be  equal.  Draw  horizontal  lines  through 
each  of  these  points;  the  distances  between  them  will  thus  represent  the  spaces 
passed  over  on  the  diameter  in  equal  periods  of  time.  Draw  also  a  series  of 
vertical  lines  equally  spaced  and  at  right  angles  to  the  first  lines,  representing 
the  distances  laterally  passed  over  in  equal  time  intervals.  Suppose  a 
particle  to  start  at  A '.  At  the  end  of  the  first  instant  of  time  the  effort  of  the 


ART.  26] 


A  FEW  PRINCIPLES  OF  OPTICS 


35 


simple  harmonic  motion  would  be  to  move  the  particle  to  B' '.  At  the  same 
time,  the  motion  of  translation  would  tend  to  move  it  to  b.  The  resultant 
of  the  two  movements  will  be  to  move  it  to  br.  At  the  end  of  the  second 
interval  of  time  the  resultant  of  the  two  forces  drawing  it  to  C  and  c  will 
move  it  to  c',  at  the  end  of  the  third  interval  to  d',  and  so  on  to  gf  where  it  has 
reached  its  point  of  minimum  velocity  and  maximum  displacement.  Beyond 
this  point  the  velocity  increases  and  the  displacement  decreases  in  the  nega- 
tive direction  until  the  particle  has  reached  the  point  sf,  the  position  of  maxi- 
mum negative  elongation.  The  next  movement  is  again  a  retrograde  move- 
ment and  the  particle  passes  to  the  position  of  rest  at  y'.  It  has  now  com- 


r      F    ( 

T)/           J 

X 

I 

• 

Cf                              rr 

/, 

./ 

/ 

\ 

,, 

i 

\ 

/ 

2 

1 

Z* 

III' 

. 

/ 

IT 

It 

\ 

/ 

• 

*\      " 

0 

\ 

7 

"' 

p 

x 

—  f- 

2 

1      S     bcdefghijklmnopqrstuvwxui 
FIG.  48. — Harmonic  curve. 

pleted  one  cycle  and  has  moved  forward  an  equal  number  of  spaces.  If  a 
line  be  drawn  through  the  successive  points,  the  resulting  curve  is  a  harmonic 
curve.  The  distance  from  a  to  y  is  called  a  wave  length;  the  distance  from 
A  to  G  is  the  amplitude. 

The  equation  of  the  harmonic  curve  may  be  obtained  by  combining  the  equations 
of  simple  harmonic  motion  and  of  uniform  rectilinear  motion.      We  found  above1 

that  d  —  r  sin  &>/, 

where  d  is  the  ordinate  or  displacement. 

Let  /  be  the  lateral  displacement  (Fig.  48),  and  v  the  velocity,  then 


Solving  for  /  and  substituting  in  (i),  we  have  d  =  r  sin  o>  - 


But  ta  =  -^j  2  therefore 


• 
d  =  r  sin  —    . 

vl 


(8a) 


foa) 


Since  T  =  time  of  a  complete  oscillation  and  v  =   the  velocity,  the  abscissa  of 
one  wave  length  (a'yr)  will  be  vT.     Let  this  value  be  represented  by\} 

vT=\.  (loa) 

Substituting  in  (ga),  we  have 

1  Eq.  i,  Art.  23. 

2  Eq.  3,  Art.  25. 


36  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  26 

•     27r^  (     } 

If  /  is  the  distance  along  the  abscissa  Ary'y  we  have 

2x1  .2x1 

When  l  =  o,  — ,-  =  o,  sin  —r-  =  o,  and  d  =  o. 

A  /. 

When  /  =  -•»  sin —p  =  sin  90°=  i,  andd  =  r. 
4  x 

A        .       27tl  .  -Oo  i     . 

When  J=-»  sin  — r-  =  sin  TT  =  sin  180  =o,  and  0  =  0. 

2  A 

When  /  =  — »  sin  — y-  =  sin  270°=  —  i,  and  d=  — r. 
4  x 

When  /  =  X,  sin  — r-  =  sin  360°  =  o,  and  J  =  o. 

A 

From  these  equations  it  may  be  seen  that  at  the  beginning  of  a  wave,  when 
there  is  no  movement  at  right  angles  to  the  simple  harmonic  motion,  the  displace- 
ment of  the  particle  is  equal  to  zero.  With  an  abscissa  of  one-fourth  of  a  wave 
length,  the  particle  has  a  displacement  equal  to  the  amplitude,  and  it  is,  conse- 
quently, at  its  maximum  in  a  positive  direction  (Fig.  48).  With  an  abscissa  of 
one-half  wave  length,  the  displacement  again  equals  zero.  At  three-fourths  wave 
length  it  is  again  equal  to  the  amplitude,  but  in  the  negative  direction,  and  the  move- 
ment has  reached  its  maximum  in  the  opposite  direction.  At  the  end  of  a  complete 
wave  length  the  particle  has  again  reached  the  position  of  rest. 

The  equation  for  velocity  at  any  moment  in  the  harmonic  curve  may  be  obtained  by 
combining  the  speed  equation  of  simple  harmonic  motion  (2).  with  the  equation  of 
uniform  rectilinear  motion. 

Substituting  values  from  equations  (4),  (8a),  and  (loa)  in  (2)  we  have: 

2xt  2x1  2x1 

cos  a  =  v  cos  -^r=  v  cos—^-  =  v  cos  — T--  (12) 


27:1  2*1 

When  /  =  o,  -  -     =o,  cos 


T  vl 

*1 

r 


2fi  2K 

When  /  =  ->  —T-  =  90  ,  cos  —  y-  =  o,  v  =  o. 

A.  A  A 

When  l  =  -2>  --=180°,  cos  -^--i,  »'=-»• 

3^       2X1  27tl 

When  /  =  —  '  —  y-=27o  ,  cos  —  r-  =  o,  v  =0. 

A  A  A 

When  /  =  X,  -^  =  360°,  cos  -2J"=i,  v'  =  v. 

That  is,  the  speed  of  the  particle  at  the  beginning  of  the  wave  is  at  its  maxi- 
mum and  is  equal  to  the  velocity  on  the  circumference.  At  one-fourth  wave  length 
it  is  zero.  It  is  again  at  its  maximum,  but  in  the  negative  direction,  at  one-half 


ART.  27] 


A  FEW  PRINCIPLES  OF  OPTICS 


37 


wave  length,  zero  at  three-fourths  wave  length,  and  at  its  maximum  in  the  positive 
direction  at  the  completion  of  the  wave.  A  curve  (Fig.  49)  constructed  with  these 
values  is  exactly  like  the  harmonic  curve,  differing  from  it  only  in  position  by  one- 
fourth  wave  length.  If  the  maximum  value  at  the  circumference  is  represented 
by  r  =  v,  and  the  curves  are  shifted  one-fourth  wave  length,  it  will  be  found  that 
the  two  coincide  exactly.  From  their  form,  these  curves  are  known  as  sine  curves. 


£ 

1 

•     i 

/ 

vZ. 

\ 

/ 

x 

: 

s 

/ 

,' 

,/ 

,' 

r', 

\ 

/ 

i 

! 

£ 

•i 

2 

' 

U'D'l 

"  1 

"   < 

/ 

^ 

/ 

\ 

\ 

/ 

^, 

\ 

/ 

i 

< 

> 

•     c 

. 

' 

! 

i 

j 

! 

/   / 

n    l 

i      i 

>     t 

'    <> 

!• 

FIG.  49. — The  velocity  curve  of  the  combination  of  simple  harmonic  motion  and  uniform  rectilinear 

motion. 

27.  Combinations  of  Simple  Harmonic  Motions. — It  was  mentioned 
above  that  a  particle  might  be  subjected  to  two  or  more  motions  at  the  same 
time.  If  two  such  motions  are  simple  harmonic  motions,  the  resulting  curve 
may  be  constructed  graphically,  or  it  may  be  calculated  by  combining  the 
equations  of  each. 

i.  Two  simple  harmonic  motions,  equal, 
along  the  same  line,  and  in  the  same  direc- 
tion. Let  OA  (Fig.  50)  be  the  amplitude  of 
a  simple  harmonic  motion.  The  first  move- 
ment would  send  the  particle  from  0  to  F'. 
The  first  movement  of  a  second  simple  har- 
monic motion  of  equal  amplitude,  acting  along 
the  same  line  and  in  the  same  direction, 
would  send  it  the  same  distance,  the  two 
together,  therefore,  sending  it  to  F".  (OF" 
=  OF'+OFr.)  The  second  movement  of  the 
first  simple  harmonic  motion  would  move  it  a 
distance  equal  to  F'E',  and  the  second  move- 
ment of  the  second  motion  would  send  it  a 
like  distance.  But  the  particle  was  already 
at  F",  consequently  the  second  movement  of 
both  simple  harmonic  motions  would  move 
the  particle  a  distance  of  2F'E'  beyond  F" 
tide  finally  reaches  a  position  at  A" 


FIG.  50. — Combination   of  two  simple  har- 
monic motions  along  the  same  line. 


or  to  E",  and  so  on  until  the  par- 
distant  20.4  from  O.  The  resulting 
motion,  therefore,  will  be  itself  a  simple  harmonic  motion  but  with  an  amplitude 
of  twice  that  of  each  of  the  original  motions. 

II.  Two  simple  harmonic  motions,  equal,  along  the  same  line,  but  in  opposite 


38 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  27 


directions.  The  first  movement  of  the  first  simple  harmonic  motion  would  tend  to 
move  the  particle  toF'  (Fig.  50),  the  first  movement  of  the  second  simple  harmonic 
motion,  to  Hf.  Since  these  motions  are  equal  and  in  opposite  phase,  the  resultant 
will  be  the  algebraic  sum  of  the  two,  or  zero.  As  these  two  movements  neutralize 
each  other,  so  will  every  other  movement,  and  the  final  result  will  be  absolute  rest. 
The  particle  will  remain  at  O. 

III.  Two  simple  harmonic  motions,  equal  but  at  right  angles  to  each  other,  and 
moving  in  the  same  phase.  Let  YY'  and  XX'  (Fig.  51)  be  two  simple  harmonic 
motions  at  right  angles  to  each  other,  and  let  A'O'B'  and.B'0"^  be  halves  of  their 
circles  of  reference.  Let  their  motions  be  equal  and  in  the  same  phase,  that  is, 
let  both  be  either  positive  or  negative.  At  the  end  of  the  first  interval  of  time, 
the  particle  0,  influenced  only  by  the  YY'  movement,  would  have  moved  to  61, 


'./ 


-1' 


ffX 


FlG.    Si. — Two    equal,    simple   harmonic 


A'  Y' 

51. — Two    equal,    simple   narmomc  FIG.  52. — Two  simple  harmonic  mo- 

motions  acting  at  right  angles  to  each  other.  tions,  equal,    at  right    angles,  and  differ- 

ing in  phase  by  one-fourth  of  a  period. 

while  if  influenced  only  by  the  XX'  movement  it  would  have  moved  to  b.  The 
actual  position  of  the  particle  will  therefore  be  at  64,  at  the  end  of  the  diagonal  of  a 
parallelogram  of  forces  whose  sides  are  Obi  and  Ob.  At  the  end  of  the  second 
interval  of  time,  the  particle  will  be  at  c4,  at  the  end  of  the  third  at  J4,  and  so  on, 
until  it  reaches  A  when  it  will  return  to  O,  and  then  move  on  to  A',  and  so  continue 
to  oscillate  between  A  and  A'  in  a  direction  at  45°  to  YY'  and  XX'.  Any  projec- 
tion of  simple  harmonic  motion  being  simple  harmonic  motion,  the  resulting  vibra- 
tion along  A  A'  is  also  simple  harmonic  motion. 

IV.  Two  simple  harmonic  motions,  equal  btit  at  right  angles  to  each  other,  and 
moving  in  opposite  phases.     Let  the  vibration  along  YY'  be  in  the  negative  direction 
and  the  vibration  along  XX'  in  the  positive  (Fig.  51).     The  resultant  of  the  first 
motion  of  the  two  simple  harmonic  motions  will  be  to  move  the  particle  to  b$,  then 
to  c5,  and  so  on  to  B  when  the  particle  will  return  and  continue  to  oscillate  between 
B  and  B'  in  a  direction  at  45°  to  YY'  and  XX'  and  at  90°  to  A  A'. 

V.  Two  simple  harmonic  motions,  equal,  and  at  right  angles  to  each  other,  but 
differing  in  phase  by  one-fourth  of  a  period.     In  Fig.  52  let  the  particle  already  have 
been  moved  by  the  YY'  simple  harmonic  motion  from  0  to  Y  when  the  OX'  compo- 


ART.  27] 


A  FEW  PRINCIPLES  OF  OPTICS 


\ 


X 


X 


x 


X 


X 


x/ 


X 


X, 


A'' 


FIG.  53. — Two  simple  harmonic  mo- 
tions, equal,  at  right  angles  to  each  other, 
and  differing  in  phase  by  less  than  one-half 


nent  begins  to  act.  That  is,  the  OX'  component  is  one-fourth  of  a  period  behind 
the  other.  Beginning  then  at  Y,  the  first  motion  of  YY'  would  tend  to  move  the 
particle  to/2  while  the  horizontal  movement  would  tend  to  move  it  to  h\,  the  result- 
ant being  a  movement  to  h.  The  second  motion  will  move  the  particle,  in  a  like 
manner,  to  i,  the  third  to  j,  and  so  on;  the  resultant  being  a  uniform  movement  in 
a  circle  in  the  Y-h-i-j-k-X'  direction.  This  clock-wise  direction  is  called  negative. 

VI.  Two  simple   harmonic  motions,  equal, 

at  right  angles  to  each  other,  but  differing  in  / 

phase  by  three-fourths  of  a  period.  Let  the 
YY'  component  already  have  made  oscilla- 
tions from  O  to  F  to  O  to  Y'  when  the  OX' 
component  starts.  The  particle  will  move 
(Fig.  52),  as  a  result  of  the  two  motions,  along 
r-q-p-o-n-X' F,  etc.,  in  a  counter  clock- 
wise or  positive  direction. 

VII.  Two  simple  harmonic  motions,  equal, 

at  right  angles  to  each  other,  and  differing  in       \ 
phase  by  less  than  one-half  a  period  but  by  some 
other  fraction   than    one-fourth.      The  ampli- 
tudes being  equal,  the  circles  of  reference  (Fig. 
53)  are  equal. 

(a)  Let  the  YY'  component  be  one-eighth    aperiod  but  by  some  other  fractionthan 
of  a  period  ahead  of  the  XX'  component,    one -fourth. 

The  particle  will,  consequently,  be  at  a  (3/24 

of  a  period  on  YY'}  when  the  XX'  motion  begins.  The  first  impulse  along 
YY'  would  move  the  particle  to  a'  while  the  first  XX'  movement  would 
move  it  to  b'",  the  resultant  being  a  movement  to  b.  The  next  impulse 
will  move  the  particle  to  c,  the  third  to  d,  and  so  on,  with  a  resulting 
curve  which  is  an  ellipse.  As  the  difference  in  phase  between  the  two 

components  becomes  greater,  the  ellipses  become  broader  (ellipse  j'V »', 

etc.)  and  finally  reach  the  circle  as  a  limiting  value  when  the  phase  dif- 
ference equals  one-fourth  of  a  period.  As  the  difference  in  phase  becomes  less,  the 
ellipses  become  narrower  (ellipse  a"b"c"  .  .  ..l"m",  etc.)  and  reach  the  limit- 
ing value  of  a  straight  line  when  the  phase  difference  equals  zero. 

(b)  If    the    difference    in    phase  is  between  one-fourth   and   one-half   of   a 
period,   the  motion  is  negative,  but  the  ellipse  has  BB'  for  its  long  diameter  in- 
stead of  A  A'. 

VIII.  Two  simple  harmonic  motions,  equal,  at  right  angles  to  each  other  but  differ- 
ing in  phase  by  some  fraction  of  a  period  other  than  three- fourths,  between  one-half 
and  a  full  period.     In  this  case  the  motion  will  be  in  the  positive  direction  as  in 
Case  VI.     (a)  With  a  difference  of  phase  between  one-half  and  three-fourths  of  a 
period,  the  ellipse  will  be  elongated  on  the  BB'  line;  (b)  with  a  difference  between 
three-fourths  and  a  whole  period,  along  the  A  A'  line. 

Eight  combinations  of  two  simple  harmonic  motions  at  right  angles  to  each 
other  have  thus  been  considered. 

i.  The  difference  of  phase  is  zero  (Case  III).  Movement  takes  place  in  the 
straight  line  A  A'  (Fig.  54). 


40 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  27 


2.  The  difference  of  phase  is  less  than  one- fourth  of  a  period  (Case  VII  a). 
The  movement  is  negative  (  — )  around  an  ellipse  elongated  on  A  A'  (Fig.  55). 

3.  The  difference  of  phase  is  one-fourth  of  a  period  (Case  V).     The  movement 
is  negative  (  — )  around  a  circle  (Fig.  56). 

4.  The  difference  of  phase  is  greater  than  one- fourth  and  less  than  one-half  of  a 
period  (Case  VII  b).     The  movement  is  negative  (  — )  around  an  ellipse  elongated 
•on  BB'  (Fig.  57). 


FIGS.  54  TO  6 1. — Directions  of  movement  in  combinations  of  two  simple  harmonic  motions. 


/" 


\ 


\ 


\rfj 


5.  The  difference  of  phase  is  one-half  a  period  (Case  IV).  The  movement  is 
in  the  straight  line  BB'  (Fig.  58). 

1  6.  The  difference  of  phase  is  greater  than  one-half  and  less  than  three-fourths 
of  a  period  (Case  VIII  a).  The  movement  is  positive  (+)  around  an  ellipse  elon- 
gated on  BB'  (Fig.  59). 

7.  The  difference  of  phase  is  three- 
fourths    of    a    period    (Case    VI).      The 
movement  is  positive  (+)  around  a  circle 
(Fig.  60). 

8.  The  difference  of  phase  is  greater 
than  three-fourths  but  less  than  a  whole 
period  (Case  VIII  b).     The  movement  is 
positive  (+)  around  an  ellipse  elongated 
on  AA'  (Fig.  61). 

When  the  difference  of  phase  is  unity, 
the  effect  is  the  same  as  in  No.  i.  Of 
course  if  the  XX'  movement  is  in  advance 
of  the  YY'j  the  motions  will  be  reversed. 
From  this  summary  it  is  clearly  evi- 
dent that  compounding  two  equal  sim- 
ple harmonic  motions  at  right  angles  to 
each  other  will  produce  elliptical  motion 
in  every  case,  limiting  values  being  the 
straight  line  when  the  phasal  difference 
is  zero  or  one-half  of  a  period,  and  the  circle  when  the  phasal  difference  is  one- 
fourth  or  three-fourths  of  a  period. 

IX.  Two  simple  harmonic  motions  at  right  angles  to  each  other,  unequal  in  ampli- 
tude but  in  the  same  phase  (Cf.  Case  III).  If  the  amplitudes  are  unequal  the  auxil- 
iary circles  will  be  of  different  size  (Fig.  62).  Let  the  two  movements  be  positive. 
Being  in  the  same  phase,  the  first  impulse  of  the  YY'  movement,  acting  alone, 
would  move  the  particle  from  0  to  63,  and  the  first  impulse  of  the  XX'  movement, 
acting  alone,  move  it  to  64.  The  resultant  of  the  two  movements  would  send 


A'  r 

PIG.  62. — Two  simple  harmonic  motions  at 
right  angles  to  each  other  and  unequal  in 
amplitude. 


ART.  28] 


A  FEW  PRINCIPLES  OF  OPTICS 


41 


it  to  65.     The  resultant  of  all  the  impulses  will  be  to  move  the  particle  to  A, 
and  it  will  oscillate  between  A  and  A'  in  a  straight  line. 

X.  Two  simple  harmonic  motions  at  right  angles  to  each  other,  unequal  in  ampli- 
tude, and  in  opposite  phase.     If  the  movements  are  in  opposite  phase,  that  is,  if 
they  differ  by  half  a  period,  the  particle  will  oscillate  between  B  and  B'  (Fig.  62. 
Cf.  Case  IV). 

Since  the  amplitudes  are  unequal  in  Cases  IX  and  X,  the  movements  along  AA' 
and  BB'  will  not  be  at  right  angles  nor  at  45°  to  XX'  and  YY'. 

XI.  Two  simple  harmonic  motions  at  right  angles  to  each  other,  of  different  ampli- 
tudes, and  differing  in  phase  by  one-fourth  of  a  period  (Cf.  Case  V).     The  movement 
is  in  the  negative  direction  as  in  Case  V,  but  here,   since  the  amplitudes  of  the 
two  motions  are  different,  the  curve  is  an  ellipse  instead  of  a  circle  (Fig.  62). 

XII.  Two  simple  harmonic  motions  at  right  angles  to  each  other,  of  different  am- 
plitudes, and  differing  in  phase  by  three-fourths  of  a  period  (Cf.  Case  VI).    The 
movement  is  in  the  positive  direction  around  an  ellipse. 

XIII.  Two  simple  harmonic  motions  at  right  angles  to  each  other,  of  different 
amplitudes,  and  differing  in  phase  by  some  other  fraction  of  a  period  than  one-fourth 
but  less  than  one-half  of  a  period  (Cf.  Case  VII).    The  movement  will  take  place  in 
the  negative  direction  around  an  ellipse. 

XIV.  Two  simple  harmonic  motions  at  right  angles  to  each  other,  of  different 
amplitudes,  and  differing  in  phase  by  some  fraction  of  a  period  other  than  three- fourths, 
and  between  one-half  and  a  full  period  (Cf.  Case  VIII).     The  movement  will  take 
place  around  an  ellipse  in  the  positive  direction. 

28.  Combinations  of  Harmonic  Curves. — We  have  already  seen  that  a 
simple  harmonic  motion  may  be  combined  with  a  uniform  rectilinear  motion 
to  give  a  harmonic  curve  (Fig.  48).  Two  harmonic  curves  in  the  same  plane 
may  likewise  be  combined,  and  the  resultant  will  be  a  different  harmonic 
curve  in  the  same  plane. 


^-  

/ 

\ 

e 

- 

\ 

- 
-' 

r 

/ 

/' 

.v 

[ 

'" 

/ 

" 

'1 

\ 

_ 

i  /<- 

/ 

/: 

^ 

•• 

' 

/ 

X 

/ 

\0 

/ 

/," 

,.,. 

•7" 

,,„ 

§ 

9 

1," 

," 

.„ 

k" 

l" 

V 

//" 

"" 

l<" 

'i" 

§ 

i 

i" 

>/" 

/•" 

«•" 

i" 

£ 

\    L 

<\ 

),' 

/ 

III 

K 

/! 

U" 

V  .S 

N 

/ 

' 

\ 

s 

/ 

i 

\, 

k 

/ 

i 

t 

II 

1C 

/ 

i 

\ 

,/ 

^ 

2 

FIG.  63. — Combination  of  two  harmonic  curves  having  the  same  amplitudes  and  wave  lengths  and 

acting  in  the  same  phase. 

I.  Two  harmonic  curves  having  the  same  amplitudes  and  wave  lengths,  and  in  the 
same  phase.  Let  Ob'c'd'  .  .  .  h'  .  .  .  etc.  (Fig.  63)  be  a  harmonic  curve.  Let 
another  harmonic  curve,  having  the  same  amplitude  and  wave  length,  and  acting 
in  the  same  phase,  also  pass  through  O.  It  likewise  will  occupy  the  position 
Ob'c'd'  .  .  .  h'  .  .  etc.  If  the  two  motions  act  together,  the  resultant  will  be  a 


42 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  28 


harmonic  curve  having  the  same  wave  length  but  an  amplitude  which  at  any  point 
is  the  algebraic  sum  of  the  two  displacements  at  that  point.  Thus  b"b'-}-b"b'  — 
b"b,  c"c'+c"c'  =  c"c,  etc. 


/ 

/', 

/ 

\ 

•" 

// 

/ 

\ 

I" 

/ 

\ 

/ 

\ 

_ 

f   • 

I 

d 

X 

I 

h 

i 

k 

\ 

L 

a 

g 

q 

\ 

/ 

t 

//• 

T  N  '// 

( 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

A 

\ 

b 

\ 

/ 

/" 

K 

\ 

/ 

" 

\ 

/ 

\ 

/ 

) 

t 

.1 

i 

>'' 

i 

FIG.  64. — Combination  of  two  harmonic  curves  of  the  same  amplitudes  and  wave  lengths  but  in 

opposite  phases. 

II.  Two  harmonic  curves  having  the  same  amplitudes  and  wave  lengths  but  in 
opposite  phase.     Two  harmonic   curves   having   the  same  amplitudes   and  wave 


/    ^L 

/ 

/ 

i 

i 

</ 

/ 

/  /^ 

\ 

\ 

/ 

\ 

\ 

/ 

f/ 

/f!\ 

\ 

\ 

/ 

/ 

/\' 

\ 

\ 

/ 

/ 

f 

/ 

\ 

e' 

V 

\ 

„ 

/ 

1 

^ 

/ 

\ 

k1 

X 

\ 

III" 

/ 

>>/ 

/ 

t 

\ 

\ 

\ 

. 

/ 

'.!' 

/" 

/ 

i" 

\ 

\ 

\ 

/ 

/ 

/ 

'!" 

„  \ 

\ 

5 

\>/ 

1 

/ 

\ 

\y 

/ 

. 

\    X^r= 

\ 

\ 

/\ 

1 

/ 

\ 

\ 

7s; 

/ 

/ 

vc^- 

e 

\ 

/ 

il 

III 

\ 

/ 

3 

•     "^^-___ 

s 

// 

\ 

FIG.  65. — Combination  of  two  harmonic  curves  of  the  same  amplitudes  and  wave  lengths  but  acting  in 

different  phases. 

lengths  but  in  opposite  phase  have  equal  opposite  displacements  at  any  point  on 
the  curve  (Fig.  64).     Being  in  opposite  phase  the  two  curves  differ  by  half  a  period 


T  '» 

FIG.  66. — Combination  of  two  harmonic  curves  of  the  same  wave  lengths  and  phase  but  differing  in 

amplitudes. 

(i/2\).  The  amplitude  at  any  point  will  be  the  algebraic  sum  of  the  two  displace- 
ments at  that  point.  Thus  at  c  the  resultant  of  cc"  and  cc'  equals  zero  since  cc' 
is  equal  tocc"  (-\-ccf  — cc"  =  o).  The  same  result  is  obtained  for  every  other  point 


ART.  28] 


.1   /••/•: II"  PRINCIPLES  OF  OPTICS 


43 


on  the  curve,  whereby  the  resultant  of  two  harmonic  curves  of  the  same  ampli- 
tudes and  wave  lengths,  but  differing  by  one-half  of  a  period,  is  zero,  or  complete 
rest.  The  curve  is  a  straight  line. 

III.  Two  harmonic  curves  having  the  same  amplitudes  and  wave  lengths,  but  differ- 
ing in  phase  by  some  fraction  of  a  period  other  than  one-half.     In  this  case  (Fig.  65) 


FIG.  67. — Combination  of  two  harmonic  curves  of  the  same  wave  length  but  differing  in  amplitudes 

and  opposite  in  phase. 

the  resulting  harmonic  curve  will  be  of  the  same  wave  length  as  either  component 
but  differ  from  them  in  amplitude.  Its  amplitude  will  be  less  than  that  in  Case  I, 
and  greater  than  that  in  Case  II. 

IV.  Two  harmonic  curves  having  the  same  wave  lengths  and  in  the  same  phase  but 
differing  in  amplitudes.  In  this  case  (Fig.  66)  the  result  obtained  by  determining 
the  algebraic  sum  at  every  point  is  a  harmonic  curve  of  the  same  wave  length  and 
phase  as  either  component,  but  differing  in  amplitude. 


\ 


\ 


FIG.  68. — Combination  of  two  harmonic  curves  of  the  same  wave  lengths  but  differing  in  phase  by 
some  fraction  of  a  period  other  than  one-half,  and  differing  in  amplitude. 

V.  Two  harmonic  curves  having  the  same  wave  lengths  but  different  amplitudes 
and  opposite  phases.     The  resultant  (Fig.  67)  is  a  harmonic  curve  of  the  same  wave 
length  but  differing  in  amplitude  from  either  component.     The  amplitude  is  the 
least  possible  of  any  combination  of  the  original  curves. 

VI.  Two  harmonic  curves  having  the  same  wave  lengths  but  differing  in  amplitude 
and  differing  in  phase  by  some  fraction  of  a  period  other  than  one-half.     In  this  case 
(Fig.  68)  the  resultant  is  of  the  same  wave  length  as  either  component  but  it  has  an 
amplitude  which  is  less  than  that  in  Case  IV  and  greater  than  that  in  Case  V. 


44 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  28 


From  these  six  cases  we  see  that  no  matter  what  the  amplitude  or  what  the 
phasal  difference,  if  the  original  components  have  equal  wave  lengths,  the  resulting 
wave  length  is  the  same.  The  amplitude,  however,  decreases  from  a  maximum  of 
the  sum  of  the  two  amplitudes,  when  there  is  no  phasal  difference,  to  a  minimum 
when  the  components  differ  by  half  a  wave  length,  this  minimum  being  the  alge- 
braic sum  of  the  two  displacements,  which,  of  course,  is  equal  to  zero  when  the  am- 
plitudes are  the  same. 

VII.  Two  harmonic  curves  of  different  wave  lengths  with  equal  or  unequal  ampli- 
tudes. The  resulting  curve  in  the  case  of  two  harmonic  curves  of  different  wave 
lengths  and  with  equal  or  unequal  amplitudes  is  much  more  complex,  and  differs 
both  in  amplitude  and  wave  length  from  either  component.  It  was  drawn  in  Fig. 
69,  as  were  all  the  preceding  curves,  by  determining  the  algebraic  sum  of  the  dis- 
placements at  different  points. 


^^ 

z 

\ 

I/ 

/    / 

\ 

\ 

\ 

/ 

// 

/ 

\ 

\\ 

/ 

(  « 

/ 

\ 

\ 

,\ 

7~ 

\ 

// 

/ 

\ 

\\ 

/ 

\ 

/ 

I 

3 

\   s 

\ 

\ 

/ 

// 

\\ 

\  / 

/ 

h 

y 

V  \ 

\ 

j 

\ 

\ 

\  / 

/ 

\\ 

v 

/ 

^^2 

V  - 

—  ^ 

^- 

2 

/ 

. 

i 

\  —  7 

/ 

/ 

FIG.  69. — Combination  of  two  harmonic  curves  of  different  wave  lengths  and  of  unequal  amplitudes. 

The  amplitude  of  the  resultant  of   two  simple  harmonic  movements  may  be 
shown  analytically  as  follows: 

We  have  as  the  equation  for  the  displacement  of  a  particle  at  any  time,1 


=  ri  sm 


and  for  a  second  vibration 


sin 


Since  the  resulting  amplitude  is  the  algebraic  sum  of  the  amplitudes  of  the 
separate  components,  we  have 


d2=:ri  sm  ------  ^ 

27T/  27tti 


=  sm 
If  we  let 


27l(t-tz) 

sm  ---------  — 


27tti  27T/2  27lt 

cos  -^-  +fj  cos  —  =  —  cos  ~>p~ 


27tt\  27lt<\ 

sm  ~T     '  r*  sm  ~~rr) 


27T/3  27T/!  27T/2 

A  cos--  =  ri  cos  -    —  \-r»  cos  -~> 


(i) 


1  Eq.  8,  Art.  25. 


ART.  28] 
and 

we  have 


A  FEW  PRINCIPLES  OF  OPTICS 


27T/3  27T/!  27T/2 

A  sin  -T  —  r\  sin  —  ,-  +r2  sm  ~~ 


. 
=  sin  -=-     A  cos  - 


27T/3\  2xt  (  .  27r/3\ 

-y/   —  cos-y^/1  sin  -=-  1 


27T/  2-/3  27T/      . 

=  ^4  sin  -r  cos  -    -  -  ,4  cos          sm 


-=-, 


whence 

d  =  A  sin 

Squaring  (i)  and  (2),  and  adding,  we  have 
.42  sin2  ~~r-\-  cos2  -^3  =  r2i4 


.        27T      /- 

A  sm 


45 


(2) 


(3) 


and 


cos 


(4) 


which  is  the  equation  of  the  amplitudes  of  the  resultant  vibration  of  two  harmonic 
motions. 

We  may  arrive  at  the  same  equation  geo- 
metrically as  follows: 

In  Fig.  70  let 

Oa  =  r\j  the  amplitude  of  the  first  vibration, 
bc  =  Oa  =  ri, 

Ob  =  r<>,  the  amplitude  of  the  second  vibration, 
ac  =  Ob  =  r«, 

Oc  =  A,  the  amplitude  of  the  resultant  of  rt 

j 
and  r2, 

Oe  =  d\y  the  displacement  of  the  first  vibration, 

Of=d2,  the  displacement  of  the  second  vibra-      FlG-  70.—  Amplitude  of  the  resultant  of 

,.  two  simple  harmonic  movements. 

tion, 
Oj=d3,  the  displacement  of  the  resultant. 

Draw  Oi,  so  that  iOX=  a=  &>/=  —• 


the  angular  displacement  of  the  first  vibration, 


~->  that  of  the  second,  and 


C0«r,.—  —  i,  that  of  the  resultant. 
Solving  the  triangles  given  in  Fig.  70,  we  have: 


46  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  28 


Substituting  values  from  those  given  above, 


27r*i  27T/i  2-irtz  .  ,  27T/2 

-^,  —  \-ir\  cos  —  ~—  •  r2  cos  —~  —  \-rzz  cosj  —  ~r 

0  27r£l    ,  ,       2irt\  27T/2    .  .      „   27T/2 

r2i  sin2—  ~  —  h2ri  sm—-  •  r%  sin  -=  —  HT  2  sin2  -~— 


-r«,  (sin*  ^+  cos*  ^)  +f  ',  (sin*  ^ 


cos*    - 

27T/!  27T^2     .  27T^1  .          27T/2 

~j~  '  cos  -^+  sm  ^y"  '  sm  ~f~ 

But  the  sum  of  the  squares  of  the  sine  and  cosine  of  an  angle  is  equal  to  unity,1 
and  the  sum  of  the  product  of  the  sines  and  cosines  of  two  angles  is  equal  to  the 
cosine  of  their  difference,2  therefore 

27T    (/2—  /l). 

cos  --    — 


This  is  the  same  equation  as  equation  (4)  above. 

GENERAL  BIBLIOGRAPHY 

1690.  C.  H.  D.  Z.  (Christiaan  Huygens  van  Zuilichem)  :  Traite  de  la  Lumiere.  Leide 
1690.  Reprint  in  German  in  Ostwald's  Klassiker  der  Exakten  Wissenschaften 
Nr.  20,  Leipzig,  2  Aufl.,  1903. 

1704.  Sir  Isaac  Newton:  Opticks.  Reprinted  in  German  in  Ostwald's  Klassiker  der 
Exakten  Wissenschaften,  Nr.  96-97. 

1815.  A.  Fresnel:  Premiere  memoire  sur  la  diffraction  de  la  lumiere,  1815. 
Idem:  Deuxieme  memoire  sur  la  diffraction  de  la  lumiere,  1815. 

1818.  Idem:  Note  sur  I'  'application  du  principe  de  Huyghens  et  de  la  theorie  des  interferences 

aux  phenomenes  de  la  reflexion  et  de  la  diffraction,  1818. 

1819.  Arago  et  Fresnel:   Memoire  sur  V  action  que  les  rayons  de  lumiere  polarisee  exercent 

les  uns  sur  les  autres.    Ann.  chim.  et  phys.,  X  (1819),  288-305.     Reprinted  in 

A.  Fresnel:  Oeuvres  completes,  I,  Paris,  1866. 
1821.  A.  Fresnel:  Explication  de  la  refraction  dans  le  systeme  des  ondes,  1821.     Also  Oeuvres 

completes,  I,  28,  117,  201,  373. 
1825.  Idem:  Ueber  das  Licht.     Pogg.   Ann.,   Ill    (1825),   89-128,   303-328;   IV   (1825), 

223-256;  XII  (1828),  197-249,  366-399. 
1832.  W.  Hamilton:  Third  supplement  to  an  essay  on  the  theory  of  systems  oj  rays.     Read, 

Jan.  23,  1832.     Trans  Roy.  Irish  Acad.,  XVII  (1837),  1-144. 
!833.  James  M'Cullagh:  Geometrical  propositions  applied  to  the  wave  theory  of  light.     Read 

June  24,  1833.     Trans.  Roy.  Irish  Acad.,  Dublin,  XVII  (1837),  241-263. 
1835.  F.  Neumann:  Theoretische  Untersuchungen  der  gesetze  nach  welchen  das  Licht  an  der 

1  Eq.  37,  Appendix. 

2  Eq.  57,  Appendix. 


ART.  28]  A  FEW  PRINCIPLES  OF  OPTICS  47 

Grenze  ziveier  vollkommen    durchsichtiger  Medien  reflektiert  und  gebrochen  wird. 

Abh.  Akad.  Wiss.,  Berlin.,  Math.  Abt.,  Pt.  I,  1835,  1-160. 
Idem:  Pogg.  Ann.,  XLII  (1837),  1-37. 
1837.  James  MacCullagh:  On  the  laws  of  crystalline  reflexion  and  refraction.     Read  Jan. 

9,  1837.    Trans.  Roy.  Irish  Acad.,  XVIII  (1839),  31-74. 
Idem:  On  the  laws  of  reflexion  from  crystallized  surfaces.     Phil.  Mag.,  VIII  (1836), 

103-108. 

Idem:  On  the  laws  of  crystalline  reflexion.     Phil.  Mag.,  X  (1837),  42-45. 
1862.  G.  G.  Stokes:  Report  on  double  refraction.     Rept.  Brit.  Asso.  Adv.  Sci.,  for  1862. 

London,  1863,  253-282. 

1885.  Th.  Liebisch:    Ueber  die   Total  reflexion   an  optisch  einaxigen   Krystallen.     Neues 

Jahrb.,  1885  (I),  245-253- 
Idem:  Ueber  die   Total  reflexion   an   doppclbrechenden  Krystallen.      Neues    Jahrb. 

1885  (II),  181-211,  1886  (II),  47-66. 
R.  T.  Glazebrook:  Report  on  optical  theories.     Rept.  Brit.  Asso.  Adv.  Sci.  for  1885, 

London,  1886,  157-261. 

1886.  Idem:  Physical  optics,  London,  2  ed.,  1886. 

1891.  Th.  Liebisch:  Physikalische  Krystallographie,  Leipzig,  1891. 

1892.  L.  Fletcher:  The  optical  indicatrix  and  the  transmission  of  light  in  crystals.     London, 

1892. 
1895.  Alfred  Daniell:  A  text-book  of  the  principles  of  physics.     New  York,  3d.  ed.,  1895. 

1900.  Henry  Crew:  The  wave  theory  of  light.     Memoirs  by  Hiiyghens,  Young  and  Fresnel. 

New  York,  1900,  81-144. 

1901.  Thomas  Preston:  The  theory  of  light,  3d  ed.,  London,  1901. 

1902.  Henry  A.  Miers:  Mineralogy.    London,  1902. 

1904.  A.  Winkelmann:  Handbuch  der  Physik,  VI,  Optik.    Leipzig,  1906. 

1904.  Rosenbusch    und    Wiilfing:  Mikroskopische   Physiographic.     Stuttgart,    4te    Aufl., 

I-i.  1904,  51-104. 

1905.  P.  Kaemmerer:  Ueber  die  Reflexion  und  Brechung  des  Lichtes  an  inactiven  durch- 

sichtigen  Kry stall platten.     Neues  Jahrb.,  B.  B.,  XX  (1905),  159-320. 

1905.  P.  Groth:  Physikalische  Kry stallo graphic.    Leipzig,  4te  Aufl.  1905. 

1906.  F.  Pockels:  Lehrbuch  der  Kristalloptik.     1906.* 

1906.  P.  Drude:  Lehrbuch  der  Optik.    Leipzig,  2te  Aufl.,  1906.     An  English  translation 
by  Mann  and  Millikan,  London,  1902. 

1906.  Joseph  P.  Iddings:  Rock  minerals.     New  York,  1906. 

1907.  Duparc  et  Pearce:  Traite  de  technique  mineralogique  et  petrographique.     Leipzig, 

1907 
1909.  Arthur  Schuster:  Theory  of  optics.     London,  2nd  ed.,  1909. 


CHAPTER  IV 
ISOTROPIC  MEDIA 

29.  Definitions. — Substances  in  which  the  velocity  of  the  transmission 
of  light  is  independent  of  the  direction  of  vibration  are  called  isotropic  (to-os, 
equal,  and  T/OOTT^  a  turning).     They  include   amorphous  substances,  such 
as  gases,  liquids,  and  annealed  glasses,  and  all   unstrained  crystals  of   the 
isometric  system. 

Substances  in  which  the  velocity  of  the  transmission  of  light  differs  in 
different  directions  are  called  anisotropic. 

30.  Wave  Motion  in  Isotropic  Media. — We  may  now  consider  the  move- 
ment of  a  series  of  particles  equally  spaced  along  a  line  in  an  isotropic  medium 
in  which  the  light  travels  with  equal  velocities  in  all  directions.     Let  a  b  c  d  e 


h 


FIG.  71. — Wave  motion  transmitted  along  a  series  of  particles    in  an    isotropic  medium. 

m,  Fig.  7 1 ,  represent  such  a  series  of  particles  in  equilibrium.     If 

some  force  displaced  the  particle  a,  for  example,  in  the  direction  a\,  the 
equilibrium  would  be  disturbed,  and  a  pull  would  be  exerted  upon  the  particle 
b  in  the  direction  bif  The  movement  of  b  would  set  up  a  movement  in  c,  and 
so  on.  In  the  meantime  the  particle  a  would  have  moved  on  in  a  direction 

at  right  angles  to  the  line  a m,  f  or  a  distance  which  was  governed  by 

the  impulse  it  originally  received  and  the  pull  exerted  by  the  other  particles. 
It  would  move  outward  with  gradually  decreasing  velocity  until  it  had 
reached  the  limit  of  displacement  at  az.  At  the  same  time  the  particle  b 
would  have  reached  £2,  and  c,  d,  while  d  would  not  yet  have  felt  the  pull. 
The  particle  a  would  now  tend  to  return  to  its  original  position  of  rest,  but 
would  be  carried  by  its  momentum  almost  an  equal  distance  on  the  other 
side  to  dQ.  Meantime  b  also  would  have  been  carried  backward,  although 
a  fraction  behind  #.  While  the  particles  first  moved  were  thus  moving  to 

48 


ART.  34]  ISOTROPIC  MEDIA  49 

the  opposite  side,  the  particles  in  advance  would  still  be  drawn  down  until 
each  had  reached  the  limit  of  its  impulse,  for  example,  g  to  #3.  The  move- 
ment of  all  the  particles  thus  vibrating  will  be  that  of  a  harmonic  curve.  It 
is  the  movement  imparted  to  a  rope,  held  at  both  ends,  when  shaken  up  and 
down.  While  each  particle  retains  its  relative  position,  a  progressive  wave 
seems  to  travel  along  the  rope.  If  only  enough  energy  is  given  to  the  line 
of  particles  to  cause  each  to  perform  a  single  oscillation,  only  a  single  wave 
travels  along  the  line,  successive  particles  having  energy  imparted  to  them 
while  the  line  behind  the  wave  sinks  to  rest.  If  the  energy  originally  imparted 
is  great  enough  so  that  the  particle  does  not  stop  at  the  end  of  a  single  oscilla- 
tion, a  succession  of  waves  of  gradually  diminishing  amplitude  travels  along 
the  cord.  If  a  continuous  periodic  force  agitates  the  line,  a  succession  of 
equal  waves  travels  along  it. 

In  a  light  wave  the  distance  between  two  particles  in  the  same  position 
and  moving  in  the  same  phase  (a*  and  m&,  Fig.  71),  is  called  the  wave  length 
and  is  represented  by  A.  The  distance  a  to  a&,  from  the  position  of  rest  to 
the  position  of  maximum  displacement,  is  called  the  amplitude. 

31.  Intensity  of  Light. — The  intensity  of  light  in  the  physical  sense,  as 
contrasted  with  the  physiological  sense,  depends  upon  the  amplitude  of  its 
vibrations.     That  is,  it  depends  upon  the  force  of  the  original  impulse:  the 
greater  the  original  displacement,  the  greater  the  intensity. 

32.  Color  of  Light. — The  color  of  light  depends,  with  certain  limitations, 
upon  its  wave  length.     Strictly  speaking,  the  rapidity  of  oscillation  governs 
color,  for  a  ray  of  a  certain  color,  passing  through  different  media,  changes  its 
velocity  of  propagation  and  proportionately  its  wave  length,  but  the  frequency 
of  oscillation  at  the  source  remains  constant,  and  therefore,  likewise,  the 
color.     It  is  the  number  of  waves  of  light  which  reach  the  eye  in  a  given 
time  that  determines  the  color  sensation.9    White  light  contains  waves  of 
all  colors  reaching  the  eye  simultaneously. 

33.  Velocity  and  Wave  Length  of  Light. — The  velocity  of  light  of  all 
colors  in  vacuo  is  the  same,  and  is  about  300,000  km.  per  second. 

The  wave  length  of  red  light  (solar  .4)  is  0.0007604  mm.  and  of  violet 
(solar  H,  calcium),  0.0003968  mm.  This  gives  about  395 Xio12  oscilla- 
tions per  second  for  the  red  and  about  757  X  io12  oscillations  for  the  violet. 

34.  Wave  Front  and  Wave  Surface. — A  ray  of  light,  traveling  in  an 
isotropic  substance,  will  travel  with  equal  ease  in  every  direction,  conse- 
quently, at  the  end  of  the  same  interval  of  time,  a  movement  arising  at  O, 
Fig.  72,  will  have  reached  the  points  a,  b,  c,  d,  e,f,  g;  all  equally  distant  from 
O.     The  wave  front,  as  it  is  called,  is  a  circle,  and  the  wave  surface,  in  space, 
is  a  sphere. 

Again,  consider  each  point  on  the  circle  as  a  new  center  of  disturbance. 

4 


50 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  35 


At  the  end  of  a  unit  of  time  a  movement  of  the  ray  front  b  will  have  extended 
the  motion  to  all  points  on  the  circle  b'b'.  Likewise  the  movement  at  c 
will  reach  the  circle  cV,  and  so  on;  since  the  new  radii  are  equal,  the  new 
wave  front  will  everywhere  be  parallel  to  the  original  wave  front;  that  is, 
it  also  will  be  a  circle,  and  the  new  wave  surface  will  be  a  sphere. 


.  n                                       a 

\ 

0                                       b 

)| 

n                                     c 

)i 

n                                       d 

y 

/ 

FIG.  72. — Wave  front  of  light  in  an  iso- 
tropic  medium.  Light  originating  in  a  point. 
(Huygens*  construction.) 


FIG.  73. — Wave  front  of  light  in 
an  isotropic  medium.  Light  originat- 
ing at  infinity.  (Huygens'  construc- 
tion.) 


/ 

\ 

/ 

\ 

\ 

' 

I 

£ 
t 

2 
i 

/ 

/ 

\ 

\ 

/ 

| 

• 

I 

t 

\ 

\ 

/ 

\ 

\ 

/ 

:: 

:  ! 

: 

•'•:. 

/' 

If  the  source  of  light  is  at  an  infinite  distance,  the  rays  will  be  parallel 
(Fig.  73),  and  the  points  a,  b,  c,  d  will  be  equally  distant  from  the  source, 
consequently  the  line  abed  will  be  at  right  angles  to  the  direction  of  propaga- 
tion of  the  ray.  New  impulses  from  these  points,  at  the  end  of  a  unit 

of  time,  will  lie  in  the  circles  a',  bf,  etc., 
having  equal  radii,  consequently  the  tan- 
gents to  all  of  them  will  be  a  line  parallel 
to  abed.  In  space  the  wave  surface  will  be 
a  plane. 

35.  Reflection  of  Waves. — If  a  wave  in 
its  course  meets  an  obstacle  to  its  free 
movement,  the  particles  act  as  if  com- 
pressed; they  rebound  and  a  retrograde 
movement  takes  place  exactly  equal  to  the 
hid  it  original  in  wave  length,  period,  amplitude, 
and  phase  (Fig.  74).  This  reflected  ray  will 
appear  in  form,  though  not  in  direction,  ex- 
actly as  the  original  wave  would  have  done  had  it  been  free  to  continue  its 
course. 

The  obstacle  may  not  entirely  prevent  the  light  from  passing  through, 
but  a  part  may  be  reflected  and  a  part  transmitted.  In  this  case  the  ampli- 
tude of  the  reflected  wave  will  be  less  than  the  original;  the  wave  length, 
however,  will  remain  the  same  although  transmitted  in  the  opposite  direction. 
If  a  wave  from  an  optically  denser  medium  passes  to  a  rarer,  the  ex- 
pansion of  the  particle  on  emerging  into  the  second  medium  has  the  same 
effect,  and  a  reflected  ray  of  less  amplitude  returns  into  the  denser  medium. 


FIG.  74. — The  effect  of  an  obstacle 
in  the  path  of  a  ray.  The  solid  line  in- 
dicates the  wave  as  it  appears,  the  dot- 
ted line  as  it  would  have  appi 
been  free  to  continue  in  its  original  direc- 
tion. O  is  the  obstacle. 


ART.  35]  ISOTROPIC  MEDIA  51 

In  the  same  manner,  when  a  ray  of  light,  the  so-called  incident  ray, 
strikes  a  second  surface  at  an  angle,  a  certain  amount  passes  through  and  a 
certain  amount  is  reflected. 

Let  a  bundle  of  rays  of  parallel  light  originate  in  an  isotropic  medium 
at  0,  O',  and  O",  Fig.  75.  When  the  ray  O  has  reached  the  point  a,  the  ray 
O'will  have  reached  a',  O"  will  have  reached  a" ',  and  the  ray  front  a  a'  a" 
will  be  at  right  angles  to  the  direction  of  propagation. 

The  ray  O"  will  continue  after  reaching  the  point  a",  and  will -soon 
reach  c.  At  the  same  time  the  ray  O  will  have  been  reflected  at  a.  Traveling 
in  the  original  medium,  its  velocity  will 
be  unchanged,  and  it  will  travel,  in  the 
time  that  the  O"  ray  travels  from  a"  to  c, 
a  distance  from  a  equal  to  a"c,  (ac"  =  a"c}. 
The  wave  front  of  the  ac"  ray  will  be  a 
sphere  with  a  as  its  center  and  with  a  radius 

equal  tO  a" C.  FlG-       75- — Huygens'      construction 

T       ,!  .1  ,^/  *          7      showing  the  course  of  reflected  rays  in 

In  the  same  way,  the  ray  O'  reaches  b    an  isotropic  medium. 
when  0"  is  at  bf .     O'  is  reflected,  and  its 

wave  front  is  a  sphere  with  a  radius  equal  to  b'c  (bc'  =  b'c).  The  wave 
front  of  all  the  rays  between  O  and  O"  will  lie,  when  O"  has  just  reached 
c ,  on  a  line  through  c  and  tangent  to  all  the  circles  representing  the  new  wave 
fronts. 

In  Fig.  75  the  angle  OaX'  =  a"ca,  since  Oa  and  O"c  are  parallel.  The 
angle  ac"c  is  a  right  angle,  since  c"c  is  tangent  at  c"  to  the  circle  of  which 
ac"  is  the  radius.  In  the  two  right  angle  triangles  ac"ca  and  aa"ca,  the 
line  ac  is  common  to  both,  and  ac"  =  a"c  by  construction.  Having  two 
lines  equal  in  a  right  angle  triangle,  the  angles  must  be  equal,  and 

a"ac  =  c"ca.  (i) 

Since  OaX'  =  a"ca,  the  complementary  angles  are  equal  and 

OaN  =  a"ac.  (2) 

Combining  (i)  and  (2)  we  have 

OaN  =  c"ca.  (3) 

But  c"ca+c"ac  =  ()o0,  and  Nac"+c"ac  =  c>o0,  therefore  c"ca  =  Nac".  Sub- 
stituting in  (3) 

OaN  =  Nac".  (4) 

The  angle  OaN  is  called  the  angle  of  incidence,  and  the  angle  Nac"  the 
angle  of  reflection.  Equation  (4)  therefore  means  that  the  angle  of  reflection 
is  always  equal  to  the  angle  of  incidence. 

As  the  angle  of  incidence  increases,  so  does  the  intensity  of  the  reflected 
light,  depending  also,  of  course,  upon  the  nature  of  the  reflecting  medium. 


52 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  36 


36.  Passage  of  Light  into  a  Medium  of  Different  Density. — The  particles 
of  which  an  optically  dense  medium  is  composed  may  be  considered  as  being 
more  closely  spaced  than  those  in  one  that  is  rarer,  as  is  shown  along  the  hori- 
zontal line  X'X,  Fig.  76.  A  wave  traveling  from  X'  to  X  arrives  at  m  where 
it  enters  the  optically  denser  medium.  The  particles  in  the  second  medium 
must  necessarily  move  in  unison  with  those  in  the  first,  therefore  the  period 
and  the  phase  remain  unchanged.  Since  the  second  medium  is  optically 


\ 


FIG.  76.  —  A  wave  passing  from  a  rarer  to  a  denser  isotropic  medium. 

denser  than  the  first,  a  wave  cannot  travel  so  rapidly  in  it,  for  in  the  time 
that  a  wave  can  travel  from  a  to  m  in  the  rarer  medium,  it  can  only  travel 
from  m  to  y  in  the  one  which  is  denser,  consequently  the  wave  length  must 
be  less.  The  ease  of  vibration  also  is  less  in  the  second  medium,  whereby 

the  amplitude  of  vibration  will  be  less,  a 
decrease  made  still  greater  by  the  fact 
that  a  certain  amount  of  energy  is  ex- 
pended in  producing  the  reflected  ray. 

Upon  passing  into  a  rarer  medium, 
the  particles  may  be  considered  as  being 
less  closely  spaced,  and  the  reverse  of  the 
above  takes  place. 


FIG.  77.-Huy8ens'  construction  show- 

ing  the  refraction  of  a  ray  of  light  upon 
passing  into  a  medium  of  greater  density. 


37-  Refraction  of  Light  upon  Passing 

fafo    Q^     JsotTOplC     MedlUHl    Of    Different 

Density.  —  When  light  falls  upon  the  sur- 
face of  a  transparent  medium,  a  part  of  it  is  reflected  back  into  the  first 
medium  and  a  part  passes  into  the  second,  generally  in  a  changed  direc- 
tion. The  second  part  is  said  to  be  refracted. 

Let  the  rays  of  light  O,  O',  and  O",  Fig.  77,  pass  from  air  into  a  denser 
isotropic  medium  X'X.  At  the  instant  that  the  ray  O  is  at  the  point  a, 
the  ray  O"  will  be  at  a".  This  second  ray  continues  on  to  c".  Meanwhile 
the  ray  O  has  been  partly  reflected  back  into  the  first  medium  and  partly 
refracted  into  the  second.  If  the  latter  medium  is  denser  than  the  first,  the 
distance  traveled  in  it  by  the  ray,  in  a  unit  of  time,  will  not  be  so  great. 
Let  v  =  velocity  of  light  in  air, 

v'  —  velocity  of  light  in  the  second  medium. 


ART.  37]  ISOTROPIC  MEDIA  53 

When  the  second  medium  is  denser  than  the  first,  v>v'. 
Let  /  =  time  of  transmission  of  light  from  a"  to  c" ',  or  from  a  to  c,  then 
a"c"  =  vt,  and  ac  =  v't, 

a"c"     i>t     v 
whereby  —  =^=-,.  (i) 

The  ray  front  of  the  ray  O,  at  the  instant  that  the  ray  O"  reaches  c", 
will  be  somewhere  on  the  surface  of  a  sphere  having  a  as  a  center  and  a  radius 
of  ac. 

Likewise  a  second  ray,  as  O',  will  travel  from  bf  to  c'  while  the  ray  O" 
travels  from  b"  to  c",  and  we  have  again: 


b'c'  ~V 

with  a  ray  front  somewhere  on  a  sphere  with  b'  as  a  center  and  b'c'  as  a 
radius.  The  wave  front  of  all  rays  between  O  and  O",  at  the  instant  the  ray 
O"  reaches  c" ,  will  be  a  plane  passing  through  c"  and  tangent  to  all  the  spheres 
upon  which  the  individual  ray  fronts  lie.  If,  then,  a  line  be  drawn  tangent 
to  all  these  circles  and  passing  through  c",  it  will  represent  the  trace  of  the 
wave  front.  Since  a  tangent  forms  a  right  angle  with  a  radius,  the  lines 
perpendicular  to  this  tangent  and  passing  through  a,  £>',  etc.,  will  represent 
the  direction  of  the  individual  rays.  Further,  in  Fig.  77,  aa"c"  is  a  right 
angle  by  construction,  and  ace"  is  a  right  angle  because  it  is  formed  by  a 
radius  and  a  tangent.  Therefore 

""c" 

ac'"  « 


ac 
Combining  (2)  and  (3)  we  have 


(3) 


sin  a"ac"      ac"      a"c" 

'4) 


ac         ac 


But  by  equation  (i)  we  have 


a"c"     v 
^T  =->=»,  a  constant,  therefore 


sin  a"acn     v 

=  n.  (3) 


Let  the  line  YY'  be  normal  to  XX' ',  then 
a"ac"+Yaa"  =  g 


54 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  38 


a"ac"  —  Oay  =  i,  the  angle  of  incidence.  (6) 


c"ac+ac"c 


But 
whereby 

We  also  have 


whereby  ac"c  =  Y'ac  =  r,  the  angle  of  refraction. 

Substituting  (6)  and  (7)  in  (5), 

sin  i  _  v 

sin  r~v'~~n' 


(7) 


(8) 


That  is,  the  ratio  of  the  sine  of  the  angle  of  incidence  to  the  sine  of  the  angle 
of  refraction  is  constant  and  bears  the  same  ratio  as  the  respective  velocities  of 
light  in  the  two  media.  This  is  known  as  Snell's  law,  having  been  discovered 
by  Willebrod  Snellius,  professor  of  mathematics  at  Leyden,  about  1621.  It 
was  first  published  by  Descartes,  Snell  having  died  in  1626  without  having 
made  the  statement  in  print. 

38.  Index  of  Refraction. — The  definite  ratio  between  the  sines  of   the 
angles  of  incidence  and  of  refraction  of  two  substances1  is  called  the  index 
of  refraction.     It  is  necessary  that  some  medium  be  chosen  as  a  standard 
for  comparison,  air  being  the  one  generally  used,  and  the  ratio  is  then  that 
of  the  sine  of  the  angle  of  incidence  in  air  to  the  sine  of  the  angle  of  refraction 
in  the  other  medium. 

In  isotropic  media,  in  which  the  velocity  of 
light  is  the  same  in  every  direction,  this  index  of 
refraction  has  a  characteristic,  constant  value  for 
every  substance.  In  anisotropic  media  it  varies 
with  the  direction  of  transmission  and  the  char- 
acter of  the  polarization,  but  it  is  constant  for  any 
definite  direction. 

For  very  accurate  measurements  it  is  neces- 
sary to  use  monochromatic  light,  since  white  light, 
which  contains  many  constituent  rays,  is  variously 
refracted,  as  may  be  seen  in  the  spectrum.  This  difference  in  refraction 
depends  upon  the  wave  lengths  of  the  rays,  which,  in  turn,  produce  different 
colors.  Thus,  in  glass,  the  index  of  refraction  for  blue  is  greater  than  for 
red  (nv  >  np),  consequently  blue  is  most  refracted  and  red  J east,  and  the  angle 
of  refraction  for  blue  is  less  than  that  for  red.  This  difference  in  refraction 
is  called  the  dispersion  of  light  (Fig.  78). 

39.  Passage  of  Light  into  Different  Isotropic  Media. — By  trigonometry 


V    P 

FIG.  78. — Dispersion  of  light 
in  isotropic  media. 


we  have  (Fig.  79) :  sin  A  =   • 

c 

increases.     When  A  =  o,  sin  A 
1  Art.  37,  supra. 


As  the  angle  increases,  the  value  of  the  sine 
=  o;  when  A  =^90°,  sin  A  =  i. 


ART.  39] 


ISOTROPIC  MEDIA 


oo 


The  denser  the  medium,  the  less  the  velocity  of  the  transmission  of  light 
within  it.  Consequently  we  may  have  three  cases. 

a.  Media  of  the  Same  Densities  (Fig.  80). — The  velocity  of  the  light  in 
the  two  media  is  the  same,  therefore 

v  =  v',  vt  =  v't,  and  a"c"  =  ac, 


sin  i      ac 


sin  r 


ac 
ac" 


Therefore  sin  *  =  sin  r,  and  i  =  r. 

7  .  FIG.  79-— Tngonomet- 

That  is,  when  light  passes  from  one  medium  into  ric  functions, 

another  of  like  density,  there  is  no  change  in  the  direc- 
tion of  the  ray. 

b.  Rarer  to  Denser  Medium   (Fig.   81).     In  this  case  v>v',   therefore 
*"c">ae. 


sin  i 
sin  r 


ac 
ac 


ac 


Y 

FIG.  80.  FIG.  81. 

FIG.  80.  —  Refraction  of  light  on  passing  into  a  medium  of  equal  density. 
FIG.  8  1.  —  Refraction  of  light  on  passing  from  a  rarer  to  a  denser  medium. 
FIG.  82.  —  Refraction  of  light  on  passing  from  a  denser  to  a  rarer  medium. 


PIG.  82. 


Therefore  sin  *>sin  r,  and  i>r. 

That  is,  when  light  passes  from  a  rarer  to  a  denser  medium,  the  ray  is 
bent  toward  the  normal. 

c.  Denser  to  rarer  medium  (Fig.  82).—  In  this  case  v'<v',  therefore 
a"c"<ac. 


sm  r      ac 


Therefore  sin  i<sin  r,  and  i<r. 

That  is,  when  light  passes  from  a  denser  to  a  rarer  medium,  the  ray  is 
bent  away  from  the  normal. 


56 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  40 


40.  Relation  between  Indices  of  Refraction  and  Velocity  of  Propagation 
of  Light. — From  equation  (8),  Article  37,  we  have 


sin  i     i) 

•-.- —  =  -f 
sin  r     v 


If  we  compare  two  substances  whose  indices  are  ri  and  nn ',  we  have, 
Combining,  we  have 


n'=  7  and  n" '  =  —,- 

1)  V 


n       v       v 
n"      v       v 

V 


// 


That  is,  for  a  given  color  of  light,  the  indices  of  refraction  of  two  media 
are  inversely  proportional  to  the  relative  velocities  of  the  propagation  of  light 
within  them.  If  one  medium  be  taken  as  a  standard  and  considered  unity, 

(n"  and  v"=i)  the  equation  becomes  n  —  —' 


41.  Total  Reflection  and  the  Critical  Angle.  —  We  found1  that       —  =  n, 

sin  / 

where  i  is  the  angle  of  incidence  in  air.  If  we  reverse  the  course  of  the  light 
and  let  it  pass  from  the  denser  to  the  rarer  medium,  and,  to  avoid  confusion, 
substitute  A  for  i  and  M  for  r,  we  have 


Rarer 
medium 


Denser 
medium 


.Limiting  position 


C.A 


0 
FIGS.  83  AND  84. — The  critical' angle  (M). 

sin  Ais  ,  sin  M *"  i 

-7 — 77  =  w,  and  -7 — /  =  -• 
sin  MY-  sin  AL    n 

That  is,  we  now  have  M  as  the  angle  of  the  incident  ray  in  the  denser 
medium,  and  A  as  the  angle  at  which  it  is  refracted  in  air.  If  A  =  o°,  sin  A  =  o, 
and 


sn 
sin 


n 


whereby  sin  M  —  o  and  M  =  o°. 
1  Art.  37,  supra. 


ART.  42] 


ISOTROPIC  MEDIA 


57 


Expressed  in  words,  when  the  light  falls  normal  to  a  second  medium  it 
passes  through  without  change  of  direction. 

If  .4=90°,  sin  A  —  it  and  sin  M  =  -  =  a  constant   (Fig.  83).     That  is, 

at  some  angle,  constant  for  the  same  substances,  the  refracted  ray  is  parallel 
to  the  separating  surface.  In  this  position  the  angle  of  incidence  is  called 
the  critical  angle  and  is  that  angle  whose  sine  is  the  reciprocal  of  the  index  of 
refraction. 

From  these  two  statements  it  is  clear  that  rays  of  light  which  pass  from 
a  denser  to  a  rarer  medium  at  any  angle  of  incidence  between  o°  and  the 
critical  angle,  will  be  partly  refracted  into  the  second  medium.  Light  falling 
upon  the  second  medium  at  angles  greater  than  the  critical  angle  will  be 
totally  reflected  (Figs.  83-84). 

If  the  critical  angle  of  a  substance  is  measured,  the  index  of  refraction 
of  that  substance  may  be  determined  from  the  above  formula.  Thus  for 

water  the  critical  angle  is  48°  31'. 


Sin  48°  31'  =  0.749  = 


=  -'    and 


1-335 
Sin  24°  25^  =  0.413  = 


«  =  1.335.     The  critical  angle  for  diamond  is  24°  2\ 

i          i 

—  =->  and  ^  =  2.419. 
2.419     n 

Since  the  brilliancy  of  a  mineral  depends  upon  the  amount  of  light  which 
is  reflected  from  it,  the  smaller  the  critical  angle,  the  more  totally  reflected 
light  appears,  and  the  greater  is  the  brilliancy. 

42.  Polarization,  and  Light  Polarized  by  Reflection. — We  have  said  that, 
in  general,  in  an  isotropic  medium  light  vibrates  in  all  directions  at  right 


FIG.  85. — Section  through  a  reflected  and 
a  refracted  ray  in  an  isotropic  medium. 


FIG.  86. — The  angle  of  polarization. 


angles  to  the  direction  of  propagation,  and  that  when  it  meets  with  another 
isotropic  medium  at  an  angle,  part  of  the  light  is  reflected  and  part  refracted. 
It  has  been  found  that  after  reflection  or  refraction,  the  vibrations  do  not 
move  with  the  same  freedom  in  every  direction  as  before,  but  that  they  are 
more  or  less  limited  to  two  planes,  so  that  the  ray  A  (Fig.  85),  originally 
vibrating  in  all  directions,  vibrates,  in  the  reflected  ray  L,  parallel  to  the  plane 
of  the  reflecting  surface  (in  the  figure,  perpendicular  to  the  plane  of  the  paper), 


58  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  43 

and  in  the  refracted  ray  R,  in  a  plane  at  right  angles  to  it.     The  light,  in  such 
cases,  is  said  to  be  plane  polarized. 

The  plane  of  polarization,  for  light  polarized  by  reflection,  is  defined  as 
the  plane  containing  the  incident  and  the  reflected  rays,  the  vibrations  taking 
place  at  right  angles  to  it.  The  plane  of  polarization  of  the  refracted  ray  is 
the  plane  at  right  angles  to  the  vibration  direction,  consequently  at  right 
angles  to  the  plane  of  the  incident  and  refracted  rays. 

43.  Angle  of  Polarization. — It  has  been  found  that  when  the  angle  of 
incidence  is  such  that  the  reflected  and  the  refracted  rays  make  an  angle 
of  90°  with  each  other  (Fig.  86),  polarization  is  at  its  maximum.  This 
does  not  mean  that  all  of  the  light  is  completely  polarized,  but  that  the  amount 
decreases  in  either  direction  from  a  certain  angle.  According  to  M.  Jamin, 
only  those  substances  which  have  an  index  of  refraction  of  about  1.46  com- 
pletely polarize  light  by  reflection.  The  angle  of  incidence  at  maximum 
polarization  naturally  differs  with  substances  having  different  refractive 
indices  but,  for  each  substance,  it  possesses  a  definite  value,  called  the 
angle  of  polarization. 

In  Fig.  86,  since  the  angle  of  incidence  equals  the  angle  of  reflection, 

AOY=YOL  =  i, 

YOL+LOX  =  c>o0,  i+LOX  =  go°.  (i) 

Also  LOX+ROX  =  90°,  and  Y'OR+ROX  =  90°. 

Combining,  LOX  =  Y'OR  =  r. 

Substitute  this  value  in  (i) 

By  trigonometry,  in  a  right  triangle  (Fig.  87), 

a  b 

sin  2  =  -,  sin  r  =-» 
c  c 

a 

,,       e  sin  i      c      a 

therefore  #  =  — r —  =  -s-  =  /  • 

smr      b      $ 

c 
But  tan  i  =  T>  therefore  w  =  tan  i. 

This  is  Brewster's  law  which  may  be  stated:     The  tangent  of  the  angle 
of  polarization  is  equal  to  the  index  of  refraction  of  the  reflecting  substance. 
A  few  examples  of  the  polarizing  angles  of  different  substances  follow. 


ART.  45] 


ISOTROPIC  MEDIA 


59 


PIG.  87. — Relations 
between  sine,  cosine, 
and  tangent. 


Crown  glass,  mean  index  #=1.515,  tan  2  =  1.515,  i  =  $6°3s' 
Flint  glass,  n=  1.622,          2  =  58°  21' 
Water,         ^=1.335,          i=  53°  10' 
Diamond,    ^  =  2.419,          ^  =  67°  32' 
Spinel,         ^=1.718,          i=^&&°  l°'~^^  * 

Since  the  refractive  indices  in  a  medium  differ  slightly  for  different 
colored  rays,  so  also  must  the  angles  of  polarization  differ.  If  the  index  for 
any  color  in  a  given  medium  is  known,  the  angle  of 
polarization  for  that  color  may  be  computed  from  the 
formula. 

If  the  incident  light  falls  upon  a  plate  at  some  angle 
ether  than  the  angle  of  polarization,  only  part  of  the 
light  is  polarized,  the  amount  depending  upon  the  angle; 
the  nearer  to  the  polarizing  angle,  the  greater  the 
amount.  The  remainder  of  the  ray  is  reflected  as  ordi- 
nary light,  but  if  it  is  reflected  subsequently  one  or  more  times,  the  pro- 
portion polarized  is  increased.  It  is  customary,  in  practice,  in  order  to 
get  a  strong  ray,  to  use  ten  or  twelve  parallel  thin  glass  plates,  termed  a 
pile  of  plates. 

That  light  is  polarized  when  reflected  may  be  shown  experimentally  by 
the  use  of  two  reflecting  surfaces.  A  simple  contrivance 
to  demonstrate  this  is  the  Norremberg  polarizer,  shown  in 
Fig.  88. 

44.  Variation  in  Intensity — Malus'  Law. — Malus  found 
that  the  intensity  of  light,  polarized  by  reflection  from  one 
mirror  and  reflected  from  a  second,  varies  as  the  square  of  the 
cosine  of  the  angle  between  the  two  planes  of  incidence. 

Let  a=  this  angle, 

a  =  the  value  of  the  maximum  intensity  of  light, 
/=  the  intensity  at  the  angle  a. 
Then  by  Malus'  law 


=  a  cos    a 


When 


or  180°,  cos  a=i,  cos2  a=i   and  /  =  a  or  the  maxi- 


FIG.    88.— Xor- 
remberg    polarizer.        mum    intensity. 

(SteegundReuter.)      when  a=9O°  or  27O°  ?  Cos  a  =  o,  and  7  =  o  or  darkness. 

45.  Polarization  by  Refraction. — Not  only  is  the  reflected  portion  of  the 
incident  ray  polarized,  but  the  refracted  portion  is  polarized  as  well.  The 
plane  of  polarization  of  the  refracted  ray  is  at  right  angles  to  the  plane  of 
the  incident  and  refracted  rays  (Fig.  89)  and  the  vibrations  take  place  within 
the  latter  plane. 

The  vibration  direction  of  the  refracted  ray  may  be  determined  experi- 


60 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  46 


mentally  by  inspecting  the  emerging  ray  at  E  (Fig.  89)  by  means  of  an 
analyzer,  such  as  a  piece  of  tourmaline  cut  parallel  to  c,  or  a  nicol  prism  such 
as  will  be  described  later. 

46.  Arago's  Law. — As  in  the  reflected  ray,  so  also  in  the  refracted  ray  it  is  only  a 
part  of  the  light  which  is  polarized.  Its  amount  de- 
pends upon  the  angle  of  incidence;  the  nearer  this  is  to 
the  polarizing  angle,  the  greater  is  the  amount.  There 
is,  however,  a  definite  relation  between  the  amount  of 
light  polarized  in  the  reflected  and  the  refracted  rays. 
This  is  expressed  in  Arago's  law:  The  reflected  and 
the  refracted  rays  of  light,  polarized  in  planes  at  right 
angles  to  each  other  by  reflection  from  and  refraction 
through  a  transparent  medium,  each  contain  an  equal 
amount  of  polarized  light. 


FIG.  89. — Apparatus  for 
showing  directions  of  vibra- 
tion of  the  reflected  and  re- 
fracted rays. 


CHAPTER  V 
ANISOTROPIC  MEDIA 

47.  Single  Refraction  and  Double  Refraction. — We  have  seen  that  in 
isotropic  media,  light  vibrates  with  equal  ease  in  every  direction,  conse- 
quently the  wave  surface  in  such  a  medium,  at  the  end  of  any  interval  of 
time,  is  a  sphere  through  which  light  passes  in  a  single  direction,  although 
changed  from  its  original  course.  Isotropic  substances,  therefore,  are  said 
to  be  singly  refracting. 

We  have  seen  also  that  there  is  another  class  of  substances  in  which  the 
rate  of  propagation  differs  in  different  directions.  These  substances  are 
called  anisotropic.  If  a  beam  of  light,  with  equal  vibrations  in  every 
direction,  passes  from  an  isotropic  medium  into  one  which  is  antisotropic,' 
its  vibrations  no  longer  remain  the  same.  If  the  second  medium  is  denser 
than  the  first,  the  ease  of  vibration  in  it  must  everywhere  be  less,  and  one 
direction  must  be  of  greater  ease  and  one  of  less,  then  all  the  others.  It 
has  been  determined  that  the  direction  of  least  ease  lies  at  right  angles  to 
that  of  maximum  ease,  and  one  would  naturally  suppose,  since  the  wave, 
before  entering  vibrates  in  every  direction,  that  light  entering  between 
these  two  positions  would  vibrate  with  an  intermediate  ease.  This,  how- 
ever, is  not  the  case.  The  intermediate  entering  wave  is  broken  up  into 
two  waves,  and  no  more,  and  these  waves  vibrate  at  right  angles  to  each 
other  in  the  principal  sections.  In  all  anisotropic  crystals  there  is  a  third 
direction  of  vibration  at  right  angles  to  the  other  two.  Its  value,  in  uniaxial 
crystals,  is  equal  to  either  the  maximum  or  minimum  ease;  in  biaxial  crys- 
tals it  is  intermediate  between  the  other  two,  and  is  called  the  direction 
of  intermediate  ease  although,  in  value,  it  is  not  necessarily  actually  the 
mean.  These  three  principal  vibration  axes  or  axes  of  the  optical  ellipsoid, 
as  they  are  called  (formerly,  axes  of  elasticity),  form  a  system  of  rectangular 
coordinates,  so  that,  in  every  anisotropic  mineral  section,  there  are  two 
vibration  directions  at  right  angles  to  each  other,  one  of  which  usually 
will  be  of  greater  ease  than  the  other,  although  the  greater  ease  in  a  sec- 
tion will  not  necessarily  be  the  direction  of  greatest  ease  in  the  mineral. 
This  property  of  anisotropic  crystals  of  resolving  light  rays  into  two  sets 
of  vibrations  is  called  double  refraction  or  birefringence. 

The  axes  of  the  optical  ellipsoid  have  a  definite  direction  in  a  given 
crystal,  and  the  relative  ease  of  vibration  along  any  crystallographic  axis  is 
constant  for  that  substance.  That  is,  the  vibrations  take  place  in  directions 

61 


62 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  48 


which  always  bear  the  same  definite  relations  to  the  crystallographic  axes. 
According  to  the  positions  of  their  vibration  axes,   crystals  may  be 
divided,  as  we  shall  see  later,  as  follows: 


Isotropic  crystals isometric 

f  tetragonal  .  . 

uniaxial     <   , 

[  hexagonal 

Anisotropic 


crystals 


biaxial 


hexagonal 
trigonal 

orthorhombic 

monoclinic 

triclinic    


extinction  parallel. 


extinction  inclined. 


Before  discussing  further  these  subdivisions  of  the  crystal  systems,  let 
us  see  what  takes  place  when  a  ray  of  light  passes  through  an  anisotropic 
substance. 


OPTICALLY  UNIAXIAL  CRYSTALS 

48.  Double  Refraction  in  Calcite.1 — The  divergence  of  the  two  refracted 
rays,  in  a  clear,  transparent  mineral  with  strong  double  refraction,  is  so  great 
that  an  image  viewed  through  it  appears  double  (Figs.  90-91).  This  property 


FIG.  90. 


FIGS.  90  AND  91.- 


FIG.  91. 
-Double  refraction  in  calcite. 


was  first  discovered  in  Iceland  spar  by  Erasmus  Bartholinus  in  1669,  and 
can  be  well  demonstrated  by  the  apparatus2  shown  in  Fig.  92. 

The  cleavage  angle  of  a  rhombohedron  of  calcite  (Fig.  93)  is  74°  56', 
and  the  axis  c  connects  the  obtuse  angles  of  the  faces.  If  such  a  rhombohe- 
dron is  placed  with  the  short  diagonal  of  one  of  its  faces  vertical,  it  will 
appear,  in  section,  as  shown  in  Fig.  94.  In  Fig.  92  two  such  rhombohedrons 

1  For  a  theoretical  discussion  see  R.  T.  Glazebrook:  Double  refraction  and  dispersion 
in  Iceland  spar;  an  experimental  investigation  with  a  comparison  with  Huyghen's  construction 
for  the  extraordinary  wave.     Phil.  Trans.  Roy.  Soc.,  London,  II  (1880),  421-449.     See  also 
Charles  S.  Hastings:  On  the  law  of  double  refraction  in  Iceland  spar.     Amer.  Jour.  Sci., 
XXXV  (1888),  60-73. 

2  C.  Leiss:  Die  optischen  Instrumente  del  Firma  R.  Fuess.    Leipzig,  1899,  152. 


ART.  48] 


ANISOTROPIC  MEDIA 


63 


of  Iceland  spar  are  shown,  the  one  in  the  center  (Rh)  so  mounted  that  it 
may  be  rotated  in  a  plane  at  right  angles  to  a  ray  of  light  passing  through 
the  screen  at  the  left. 


fit) 


:7~     "-  ^ ".      -.:-•  -        ^  ; 

.:::..-... 

FIG.  92. — Apparatus  for  showing  double  refraction  in  calcite.     1/5  natural  size.     (Fuess.) 

If,  now,  a  ray  of  light  (P,  Fig.  94)  passes  through  the  aperture  in  the 
screen  and  falls  upon  the  prism  at  right  angles  to  its  face,  it  will  be  found, 


FIG.  93. — A  rhombohedron  of  calcite. 


FIG.  94. — Separation  of  rays  in  calcite. 
Section  cut  parallel  to  the  c  axis. 


when  viewed  from  the  back,  that  it  has  been  broken  up  into  two.     Instead 
of  the  single  spot  of  light  which  would  have  appeared  through  an  isotropic 


FIG.  95.  FIG.  96.  FIG.  97.  FIG.  98. 

FIGS.  95  TO  98. — Positions  of  the  spots  of  light  on  rotating  a  rhombohedron  of  calcite. 

medium,  there  will  be  two  equally  bright  spots,  the  one  vertically  above 
the  other  (Fig.  95).     If  the  rhombohedron  Rh  (Fig.  92)  be  rotated,  it  will 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  48 


be  found  that  one  image  (O,  Figs.  95-98)  remains  stationary  while  the 
other  moves  around  it.  It  is  perfectly  clear  that  the  ray  O  has  passed 
through  without  changing  its  direction,  just  as  it  would  have  done  had  the 
medium  been  isotropic.  It  is,  therefore,  called  the  ordinary  ray.  The  ray 
£,  however,  acts  in  a  different  manner,  for  although  the  incident  light  falls 
normal  to  the  surface  of  the  rhombohedron,  it  is  re- 
fracted, as  shown  in  Fig.  94.  It  is  called  the  extraordi- 
nary ray. 

If  an  opaque  card,  through  which  a  very  small 
hole  has  been  punched,  is  placed  in  contact  with  the " 
farther  side  of  a  calcite  rhombohedron,  and  a  second 
card  is  placed  on  the  near  side,  it  will  be  found  that 
F  i  G.  9  9.— Experiment    there  are  two  positions  of  the  eye  where  the  image 

showing    that   light    passes  . 

through  a  calcite    crystal    can  be  seen  (Fig.  99).     Evidently  the  ray  must  have 

sTnte  Hn7  SPCedS  al°ng  the    travelec*  alonS  tne  line  ab  with  two  different  speeds, 
since    they   were    differently    refracted    when    they 

emerged  in  air.  It  will  be  found  that  the  ray  having  the  greater  velocity 
within  the  crystal  has  the  lesser  index  of  refraction,  and  vice  versa.  For 
calcite,  in  which  the  velocity  of  the  extraordinary  ray  is  greater  than 
that  of  the  ordinary  (E>O),  the  refractive  index  of  the  former  is  less  than 
that  of  the  latter  (e<aj),1  consequently,  since  it  passes  from  a  denser  to  a 
rarer  medium,  the  angle  of  refraction  is  greater. 

If,  in  the  first  experiment,  the  rhombohedron  Rhi  (Fig.  92)  be  used  in- 
stead of  the  Rh  rhombohedron,  and  it  is  rotated  until  the  apices  of  the  obtuse 
angles  between  the  faces  lie  in  the  horizontal  line — that  is,  until  crystallo- 
graphic c  is  horizontal — it  will  be  found 
that  but  a  single  spot  of  light  appears, 
just  as  it  would  do  were  the  medium 
isotropic.  If  the  rhombohedron  is  now 
rotated  on  crystallographic  c  as  an  axis, 
it  will  be  found  that  the  position  of  the 
spot  does  not  change.  There  is,  here, 
no  double  refraction,  and  to  this  direc- 
tion the  name  optic  axis,  axis  of  isotropy,  or  axis  of  no  double  refraction 
has  been  given.  If  the  rhombohedron  be  rotated  in  any  direction  other  than 
about  this  axis,  the  distance  between  the  refracted  images  will  be  found  to 
increase  to  a  maximum  and  then  again  decrease  until  the  crystal  has  been 
rotated  90°  from  its  former  position,  that  is,  until  crystallographic  c  is  ver- 
tical. In  this  position  again  but  one  spot  appears,  and  the  crystal  may  be 
rotated  about  the  vertical  axis  with  no  apparent  change.  Although  there  is 
no  bending  apart  of  the  rays  in  this  direction,  a  change  has,  however,  taken 
place,  and  one  ray  has  been  greatly  retarded,  as  we  shall  see  presently. 

1  Cf.  Art.  40,  supra. 


FlG.  100. — Double  refraction  through  two 
calcite  rhombohedrons  of  equal  thickness  and 
in  the  same  orientation. 


ART.  48]  ANISOTROPIC  MEDIA  65 

If  the  second  rhombohedron  of  calcite  (Rhi,  Fig.  92)  is  of  the  same  thick- 
ness as  the  first,  and  the  two  are  so  placed  that  their  shorter  diagonals  are 
vertical,  consequently  having  the  orientation  of  crystallographic  c  the  same, 
two  bright  spots  of  equal  illumination,  but  separated  by  twice  the  former 
distance  (Figs.  100  and  101),  will  again  appear.  If  the  second  crystal  is  of 
a  greater  or  less  thickness  than  the  first,  the  increase  in  the  distance  between 
the  spots  will  be  greater  or  less,  the  result,  in  every  case,  being  as  though 
the  original  rhombohedron  had  been  thickened  the  amount  of  the  second. 

If,  for  convenience,  we  place  the  calcite  rhomb  Rhi  (Fig.  92)  between  the 
one  which  can  be  rotated  (Rh)  and  the  screen,  and  in  such  a  position  that 
'both  of  the  rhombohedrons  are  parallel,  and  then  rotate  one  (Rh)  slightly 
either  to  the  right  or  to  the  left,  we  shall  find  that  four  images,  unequally 
illuminated,  appear  (Fig.  102).  When  the  front  calcite  has  been  rotated  45°, 


FIG.  101.  FIG.  102.  FIG.  103.  FIG.  104.  FIG.  105. 

FIGS.   101  TO  105. — The  bright  spots  as  they  appear  upon  rotating  one  of  two   calcite  rhombohedrons 

the  four  images  will  be  equally  illuminated  (Fig.  103),  there  having  been  a 
gradual  decrease  in  the  brightness  of  the  original  spots,  and  a  gradual  in- 
crease in  the  other  two.  This  decrease  and  increase  in  brightness  continues 
as  the  second  calcite  is  rotated  still  farther  until,  when  the  90°  position  has 
been  reached,  the  first  two  spots  have  disappeared  and  only  the  other  two 
remain  (Fig.  104).  Upon  farther  rotation,  the  four  spots  again  appear,  and 
remain,  with  varying  brightness,  until  the  rhombohedron  has  been  turned 
1 80°,  when  but  a  single  one  is  left  (Fig.  105).  Its  brightness  is  double  that 
of  either  of  the  first  pair. 

Upon  an  examination  of  the  observed  phenomenon,  we  see  that  two  of 
the  spots  remain  stationary,  though  with  increasing  or  decreasing  brightness, 
while  around  each  as  a  center,  another  one  revolves.  From  the  alternate 
brightening  and  darkening  of  the  spots,  it  is  clearly  evident  that  the  light  did 
not  pass  through  the  crystals  as  ordinary  light,  vibrating  equally  in  every  direc- 
tion, for  if  it  had,  the  illumination  would  have  remained  uniform.  We  can 
explain  the  phenomenon,  however,  if  we  consider  the  vibrations  of  the  extra- 
ordinary and  ordinary  rays  as  taking  place  at  right  angles  to  each  other.1 
Not  only  the  appearances  here  seen,  but  all  the  phenomena  of  polarized  light 
as  well,  may  be  explained  by  considering  the  vibrations  of  the  extraordinary 
ray  as  taking  place  in  the  plane  containing  the  ray  of  light  and  the  shorter 

1  Examine  the  spots  by  means  of  a  nicol  prism  held  in  the  hand,  and  note  their  vibra- 
tion directions. 
5 


66 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  48 


FIG.  106. — Vibration  directions  of  light  in  pass- 
ing through  two  parallel  rhombohedrons  of  calcite 
in  the  same  orientation. 


diagonal  of  the  face  of  the  rhombohedron,  and  the  vibrations  of  the  ordinary 
ray  as  perpendicular  to  it,  or  vice  versa.  The  former  assumption,  however, 
is  the  one  commonly  used.  It  is  to  be  noted  that  this  ambiguity  is  not 
separate  and  distinct  for  each  kind  of  crystal.  The  relative  polarizations 
can  be  determined,  and  if  the  direction  of  polarization  is  assumed  to  be  in  a 
certain  direction  in  one  crystal,  it  determines  the  direction  in  another. 

Applying  the  knowledge  of  these  vibration  directions  to  the  observed 
phenomenon,  we  have,  in  the  case  of  two  parallel  rhombohedrons  with  the 
same  orientation,  vibrations  taking  place  as  shown  in  Fig.  106.     The  ray  of 
light,  originally  vibrating  in  every  direction  (R),  reaches  the  first  rhombohe- 
dron and  is  polarized  into  two  rays 
at  right  angles  to  each  other  (O  and 
£).     The  plane  of  the  paper,  con- 
taining the  ray  of  light   and  the 
shorter  diagnonal  of  the  rhombohe- 
dron, is  the  plane  of  vibration  of 
the  extraordinary  ray — the  vibra- 
tions being  represented  in  the  figure 
by  the   short   parallel   lines.     The 

vibrations  of  the  ordinary  ray  (O),  taking  place  at  right  angles  to  the  plane 
of  the  paper,  are  represented  by  dots. 

After  passing  through  the  first  calcite,  the  extraordinary  ray  is  refracted 
back  to  its  original  course,  and  the  two  rays  travel  in  parallel  directions, 
but  with  vibrations  perpendicular  to  each  other,  until  they  reach  the  second 
calcite  (B).  Consider  first  the  ordinary  ray.  At  the  point  a  there  is  a 
tendency  in  the  second  rhombohedron  to  break  the  ray  into  two  sets  of 
vibrations  perpendicular  to  each  other.  The  calcite  will  permit  a  ray  with 
horizontal  vibrations  to  pass  through  in  a  direct  line  to  O0 — the  ordinary 
component  of  the  first  ordinary  ray.  It  would  also  permit  any  vertical 
vibration  to  pass  through  in  the  refracted  direction  ac,  but  since  no  such 
component  reaches  a,  only  horizontal  vibrations  of  O  reach  the  eye. 

A  ray  of  light,  on  reaching  d,  tends  to  pass  along  the  lines  df  and  dc. 
Without  refraction,  only  horizontal  vibrations  can  pass  through  the  crystals, 
but  since  there  is  no  horizontal  component  in  the  ray  which  reaches  d,  no 
light  passes  to  c.  The  extraordinary  component  of  the  extraordinary  ray 
(Ee),  however,  can  pass  along  the  line  df,  for  its  vibrations  take  place  in  the 
plane  of  the  paper.  In  consequence  of  the  suppression  of  the  extraordinary 
component  of  the  ordinary  ray  (Oe)  and  the  ordinary  component  of  the  ex- 
traordinary (Eo),  only  two  spots  reach  the  eye,  one  at  O0  and  one  at  Ee. 
In  Figs.  107  to  1 1 8,  vibration  directions,  lines  of  refraction,  and  spots  of  light 
are  shown  as  they  occur  in  the  two  crystals  when  looking  through  them 
at  the  hole  in  the  screen  from  the  right  in  Fig.  92.  The  solid  lines  are 
refraction  directions,  construction  lines,  etc.,  in  the  rhombohedron  nearest 


ART.  48] 


ANISOTROPIC  MEDIA 


67 


the  eye  (Rhi);  the  dotted  lines,  in  the  rhombohedron  which  first  refracts 
the  light  (Rh). 

After  passing  through  the  first  calcite  only,  the  spots  appear  as  in  Fig. 
107.  (The  short  lines  within  the  small  circles  indicate  the  vibration  direc- 
tion of  the  ray  which  produced  that  spot  of  light,  and  also  show,  by  their 
number,  the  relative  intensities  of  the  illumination.)  When  the  second  cal- 


FIG.  113.         FIG.  114.         FIG.  115.          FIG.  116.         FIG.   117.  FIG.  118. 

FIGS.   107  TO  118. — Diagrams  showing  the  double  refraction  in  two  calcite  rhombohedrons,  and  the 

effect  upon  a  spot  of  light  when  one  rhombohedron  is  rotated. 

cite  rhombohedron  is  placed  before  it,  the  spots  appear  farther  apart  (Fig. 
108),  as  explained  above.  When  it  is  slightly  rotated,  four  spots  appear 
(Figs.  102  and  109),  since  each  spot  of  light,  upon  reaching  the  second  rhom- 
bohedron (Fig.  109),  acts  as  a  new  center  of  disturbance  and  is  doubly  re- 
fracted, the  amount  of  light  passing  through 
depending  upon  the  angle  which  the  vibra- 
tion plane  of  the  on-coming  ray  makes  with 
the  vibration  planes  of  the  second  calcite. 
Let  CO  (Fig.  119)  represent  half  the  ampli- 
tude of  the  on-coming  ordinary  ray,  and  CE 
half  the  amplitude  of  the  extraordinary  ray. 
To  pass  through  the  second  calcite,  the  light 
must  vibrate  along  the  new  vibration  planes 
CEe  and  CO0.  To  determine  the  amplitude 
of  each  of  the  new  vibrations,  we  may  resolve 
each  of  the  original  rays,  by  means  of  the  usual  parallelogram  of  forces,  into 
two  others  acting  at  right  angles  to  each  other  and  along  the  new  axes. 
Thus  the  ordinary  ray,  with  half  its  amplitude  of  vibration  equal  to  CO, 
will  be  broken  up,  in  the  second  rhombohedron,  into  two  rays  with  half 
amplitudes  of  COe  and  CO0)  and  with  vibrations  parallel  respectively  to  the 


FIG.   119. — Resolution  of  vibrations. 


68 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  48 


short  diagonal  of  the  rhombohedron  and  at  right  angles  to  it.  The  extra- 
ordinary ray,  having  an  original  vibration  of  twice  CE,  will  be  resolved  into 
vibrations  having  half  amplitudes  of  CEe  and  CE0. 

When  the  calcite  rhombohedron  has  been  rotated  until  its  diagonals  make 
angles  of  45°  with  the  diagonal  of  the  first  rhombohedron  (Fig.  no),  the 


iiiiiiminimnD 


Fig.   120.  Pig.    121.  Fig.    122. 

FIGS.   1 20  TO  122. — Amplitudes  of  resultant  vibrations  when  the  calcite  rhombohedrons  are  rotated  to 

various  angles. 

four  spots  will  appear  of  uniform  illumination  if  the  vibration  amplitudes 
of  the  original  rays  were  equal,  for  the  amplitudes  of  the  resulting  vibrations 
will  also  be  equal  (Fig.  120). 

On  still  farther  rotation,  the  spots  which  were  brightest  in  the  positions 
shown  in  Figs.  102  and  109  are  dimmest  (O0  and  Ee,  Fig.  in),  and  the  spots 

dimmest  in  Fig.  109  are  brightest 
in  Fig.  in  (Eo  and  Oe).  That 
such  should  be  the  case  may  be 
seen  from  the  diagram  of  forces, 
Fig.  121. 

When  the  revolution  has 
reached  almost  90°,  the  vibration 
amplitude  of  Oe  has  become  nearly 
as  great  as  O,  and  E0  nearly  as 

great  as  E,  while  O0  and  Ee  have  nearly  become  zero  (Fig.  122).  At  90° 
the  vibration  directions  coincide,  and  the  amplitudes  of  E0  and  Oe  are 
equal  to  O  and  E,  while  O0  and  Ee  have  become  zero,  consequently  but 
two  spots  appear  (Figs.  104  and  112). 

Continuing  the  rotation,  the  spots  which  were  brightest  previously  be- 
come darker,  and  vice  versa  (Figs.  113-115),  until  at  180°  the  E0  and  Oe 
components  are  zero  and  the  other  two  are  at  their  maxima.  In  this  case, 
since  the  refraction  of  Ee  is  to  a  position  above  the  original  O,  and  O0  passes 
straight  through  at  the  same  point,  but  one  spot  appears  (Figs.  105  and  116). 
This  is  shown  better  in  the  cross-section  (Fig.  123),  where  the  vibrations 
are  indicated  by  dots  and  lines  in  the  same  manner  as  they  were  in  Fig.  106. 


FIG.  123. — Vibration  directions  of  light  passing 
through  two  parallel  rhombohedrons  of  calcite  in 
opposite  orientations. 


ART.  48] 


AMSOTROPIC  MEDIA 


69 


Experimentally  the  passage  of  the  light  in  Figs.  107,  108,  112,  and  116  may  be 
shown  by  a  model.  In  a  wooden  handle  (Fig.  126)  four  spring-brass  wires  are  set 
as  shown  at  b.  The  amplitude  of  the  vibration  of  the  ordinary  ray  is  represented 
by  the  distance  between  the  wires  O  and  O,  and  the  amplitude  of  the  vibration  of 
the  extraordinary  ray,  by  E  to  E.  Fig.  1 24  is  a  wooden  block  representing  the  first 
calcite  rhombohedron.  In  it  are  bored,  accurately  parallel,  two  holes,  O  and  0, 
perpendicular  to  the  face  of  the  rhombohedron,  and  two  holes,  E  and  E,  at  a  conven- 
tional angle  of  20°  to  30°,  representing  the  passage  of  the  extraordinary  ray;  the 


FIG.  124.  FIG.  125.  FIG.  126 

FIGS.   124  TO  126. — Apparatus  for  demonstrating  double  refraction  in  two  calcite  rhombohedrons. 

angle  depending  upon  the  thickness  of  the  model,  but  so  taken  that  the  points  of 
emergence  of  the  holes  on  the  lower  side  (E'E',  Fig.  124)  will  be  well  above  the 
emergence  of  the  O  holes.  Fig.  125  represents  the  second  rhombohedron.  In  it 
are  bored  eight  holes  (0),  perpendicular  to  the  front  face,  and  eight  inclined 
holes  (E),  in  the  positions  shown.  A  stiff  wire  through  C  (Figs.  124  and  125) 
serves  as  an  axis  of  rotation,  and,  by  keeping  the  rhombohedrons  separated  2  or 
3  in.,  the  passage  of  the  wires  through  the  different 
sets  of  holes  in  parallel,  90°  and  180°  positions  may  be 
shown  clearly,  and  the  vibration  directions  demon- 
strated. 

Another  helpful  model  may  be  made  by  construct- 
ing two  rhombohedrons  of  glass.  Within  the  first, 
upon  two  other  plates  of  glass  fastened  at  right 
angles  to  each  other,  are  painted  the  vibration  direc- 
tions and  the  directions  of  transmission  of  the  ordi~ 
nary  and  the  extraordinary  rays.  In  the  second 
rhombohedron,  other  plates  of  glass,  at  right  angles 
to  each  other,  are  fastened  in  such  a  way  that  painted 
lines  upon  them  represent  two  sets  of  ordinary 

and  extraordinary  rays,  the  distance  between  them  being  so  chosen  that  the  two 
sets  form  the  continuation  of  the  rays  from  the  first  rhombohedron.  By  rotat- 
ing the  second  rhombohedron  through  o°,  90°,  or  180°,  the  extinction  of  the 
proper  spots  will  be  brought  out  clearly. 

The  intensity  of  the  light  in  any  position  of  the  rhombohedrons  may  be  com- 
puted mathematically.  It  follows  the  same  laws  as  light  polarized  by  reflection 
(Art.  42),  consequently  Malus'  law  holds  here  also:  The  intensity  varies  as  the 
square  of  the  cosine  (or  sine)  of  the  angle  between  the  principal  sections  of  the 
two  rhombohedrons. 

If,  in  Fig.  127,  we  represent  the  amplitude  of  CE  by  E,  that  of  CE0  by  E0,  etc., 
and  consider  the  radius  of  the  circle  as  unity  (CE  =  CO=i),  then 


FIG.   127. — Intensity  of  emerg- 
ing rays. 


70  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  49 

Ee  =  E  COS  «  =  COS  a, 

E0  =  E  cos  (90°—  a)=E  sin  a=sin  a, 

O0  =  O  COS  a  =  COS  a, 

Oe  =  O  cos  (90°  —  a)=O  sin  a  =  sin  a. 
But  the  intensity  of  light  varies  as  the  square  of  the  amplitudes,  therefore 

Intensity  of  Ee  =  cos2  a, 

Intensity  of  E0  =  sin2  a, 

Intensity  of  O0  —  cos2  a, 

Intensity  of  Oe  =  sin2  a. 

From  these  equations  it  will  be  seen  that,  when  the  vibration  amplitudes  of  0 
and  E  are  equal,  there  will  be  but  two  light  intensities  for  any  angle,  that  is,  the 
intensity  of  Ee=00  and  of  E0  =  0e. 

49.  Optic  Axis  in  Uniaxial  Crystals. — We  saw,  by  rotating  a  calcite  prism, 
that  there  is  one  direction,  and  one  only,  along  which  there  is  no  double  refrac- 
tion.1   This  direction,  or  optic  axis,  as  it  is  called,  is  parallel  to  crystallo- 
graphic  c.     By  examining  various  other  minerals  we  would  find  that  every 
crystal  belonging  to  the  tetragonal  or  the  hexagonal   (including   trigonal) 
system,  has,  similarly,  but  one  axis  of  isotropy,  and  that,  in  every  case,  it  is 
parallel  to  crystallographic  c.  Crystals  of  these  systems,  consequently,  are 
called  optically  uniaxial. 

50.  Principal  Optic  Section  of  a  Uniaxial  Crystal. — The  principal  optic 
section  of  a  uniaxial  crystal  may  be  defined  as.  the  plane  which  contains  the 
axes  of  greatest  and  least  ease  of  vibration.^  Since  the  ease  of  vibration 
along  crystallographic  c  must  be  either  greater  or  less  than  along  any  other 
direction,  the  principal  optic  section  must  contain  that  axis.     In  the  plane 
at  right  angles  to  crystallographic  c  the  ease  of  vibration  is  the  same  in  every 
direction,  and  its  value  is  the  minimum  if  c  is  the  maximum,  and  vice  versa. 
Since  there  may  be  any  number  of  planes  through  crystallographic  c,  so  also 
must  there  be  innumerable  principal  optic  sections. 

V 

51.  Positive  and  Negative  Uniaxial  Crystals. — Biot2  first  recognized  the 
fact  that  uniaxial  crystals  could  be  divided  into  two  classes  according  to 
whether  the  index  of  refraction  of  the  ordinary  (&>)  or  of  the  extraordinary 
(e)  ray  is  the  greater.     For  convenience  of  description,  crystals  in  which  the 
refractive  index  of  the  ordinary  ray  is  the  greater  (co  >  e)  are  called  negative 
(-),  and  crystals  in  which  the  reverse  is  the  case  (a><e)  are  called  positive 
(+).     Thus  apatite,  with  to  =  1.638  and  €=1.634,  and  calcite  with  &>  =  1.658 
and  €=1.486   are  negative,  while  quartz,  with  o>  =  1.544  and  £=1.553,  is 
positive. 

1  Art.  48,  page  61,  supra. 

2  J.  B.  Biot:  Memoir  e  sur  la  decouverte  d'une  propriete  nouvelle  dont  jouissent  les  forces 
polarisantes  de  certains  cristaux.     Mem.   Acad.   France,  Annee  1812,  XIII   (1814),  pt. 
II,  19-30. 


ART.  52] 


ANISOTROPIC  MEDIA 


71 


Since  crystallographic  c  is  always  the  direction  of  vibration  of  the  ex- 
traordinary ray  in  uniaxial  crystals,  the  rule  may  be  stated,  that  if  crystallo- 
graphic c  is  the  direction  of  vibration  of  the  fastest  ray,  the  crystal  is  nega- 
tive; if  it  is  the  direction  of  the  slowest  ray,  it  is  positive. 

52.  Velocity  of  Any  Intermediate  Ray  in  a  Uniaxial  Crystal. — If  the 

maximum  and  minimum  indices  of  refraction  of  a  mineral,  and,  consequently, 
their  wave  velocities,  are  known,  it  is  possible  to  compute  the  index  of  refrac- 
tion and  the  ray  and  wave  velocities  in  any  other  direction. 

Since  the  velocity  of  the  ordinary  ray  is  the  same  in  every  direction,  its 


FIG.   128. — Ray  and  wave  velocities. 

index  of  refraction  is  likewise  the  same.  The  velocity  of  the  extraordinary 
ray,  however,  differs  in  different  directions,  therefore  the  index  of  refraction 
of  an  intermediate  ray  will  be  different. 

Let  Fig.  128  represent  a  section  through  the  extraordinary  ray  surface  of  a  nega- 
tive uniaxial  crystal  at  the  end  of  a  unit  of  time  (/). 
Let  r  =  the  velocity  of  the  desired  ray  (MR), 

«  =  the  angle  which  the  desired  ray  makes  with  the  c  axis. 

Then 

But  since  /  =  unity, 


\  =  rt. 
MR  =  r. 


CMR  =  a  =  direction  of  propagation  of  the  desired  ray, 
MC     =  direction  of  vibration  of  the  ordinary  ray, 
MA     =  direction  of  vibration  of  the  extraordinary  ray, 
MC     =  Ot,MA=El. 
But  /=  i,  therefore 

MC  =  0,    MA=E. 
From  the  equation  of  an  ellipse1  we  have 


1  Eq.  83,  Appendix. 


(i) 


x  y 

We  also  have  sin  a  =  -  ,  and  cos  «  =  -  , 


72  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  53 

We  also 

from  which 

*2  =  r2  sin2  «,  and  >»2  =  r2  cos2  a.  (2) 

Substituting  in  (i) 

OV2  sin2  «+£V2  cos2  a  =  02E2; 
O2F2 

r2  =  ________  ^J?  ____  /     1 

O2  sin2  a+E2  cos2  a 

TTws  w  aw  equation  giving  the  velocity  of  a  ray  making  an  angle  of  a  with  the  c  axis  of 
a  uniaxial  crystal.     It  is  to  be  noted,  however,  that  the  index  of  refraction  of  the 

ray  r  is  not  -,  as  at  first  sight  one  might  suppose,  but  is  of  a  different  value.     This 
will  be  proved  below. 

53.  Velocity  of  Any  Intermediate  Wave  in  a  Uniaxial  Crystal.  —  The  velocity 
of  the  wave  produced  by  the  extraordinary  ray  is  not  the  same  as  the  velocity  of 

the  ray  itself.  Following  Fresnel,  one  may  con- 
sider a  narrow  cone  of  rays  as  aMc  (Fig.  129). 
At  the  end  of  a  unit  of  time,  the  light  disturb- 
ance, arising  at  M,  will  have  reached  a,  b,  and 
c,  and  if  the  distance  ac  is  small  enough,  the 
line  abc  will  coincide  with  the  tangent  to  the 
ellipse.  In  ether  words  the  ray  front  will  be  tan- 
gent to  the  ellipse. 

If,  now,  instead  of  a  single  ray  of  light,  we 
FIG.  129.—  Fresnei's  figure  for  show-    consider   a   series  of    parallel   rays,   MR,  MR, 

ing   that  in  a   small   cone  of  rays  the     pi       I2g    we  ^j  find  ^  ^  m  ^  h  tfa 

ray  front  will  be  tangent  to  the  ellipse.  J 

plane  surface,  of  which  NRNR  is  the  trace,  at 

the  same  instant.  The  angle  CMR  will  represent  the  direction  of  the  refracted 
.rays,  and  MR  the  distance  traveled  by  them  in  a  unit  of  time.  The  wave  front, 
however,  has  only  traveled  from  MM  to  NN,  the  normal  M  N  representing  the 
actual  distance  through  which  it  has  moved. 

Let  MN  =  w,  the  velocity  of  the  wave  produced  by  the  rays  r,  r. 
<?   =CMN, 

ci  =  the  index  of  refraction  of  the  ray  r. 

Since  the  disturbance  produced  by  the  ray  r  results  in  forming  a  wave  whose 
velocity  is  w,  the  index  of  refraction  of  this  wave  is  the  index  of  the  ray  producing 
this  velocity,  or 

»-'  (4) 

Substituting  E  =  a  and  O  =  b  in  the  general  equation  of  the  intercepts  of  the 
tangent  to  an  ellipse,  we  have 


-.  (5) 

But  sin  v  =          and  cos  ^  =        7-  (6) 

Combining  (5)  and  (6),  we  have 


ART.  54]  ANISOTROPIC  MEDIA  73 

wx  wy 

sin  tp=  ,-,2  and  cos  <p  =  ^^ 

(ri 

E2  sin  <p  O2  cos  <p 

or  x=  and  y  =  — 

w  w 

Substituting  these  values  in  (i),  the  equation  of  the  ellipse,  we  obtain 


cos    <f>  _ 
w2  (8) 


or  sn    ?>          cos    <p  =  w. 

This  is  Me  equation  of  the  velocity  of  the  wave  front  of  the  extraordinary  ray  of  a 
uniaxial  crystal  in  a  medium  having  maximum  and  minimum  velocities  of  E  and  O. 
Substituting  in  this  equation 

=  —>  cos  <p=  —  »  w2  =  z'2+;y'2  (8a) 

we  obtain  EV2+O2/2=  (x'2+/2)2. 

This  is  the  equation  of  the  curve  of  the  oval  of  the  wave  fronts  of  a  uniaxial  crystal. 
If  we  substitute  in  (8)  the  values 


€  CO  €i 

sin2  <p   ,  cos2  <?      i 
we  have  ^      +    ~t     =  ^  (9) 


-      .  Y 

V  w  sm2  ^+€2  cos2  <f> 


This  is  /Ae  equation  of  the  index  of  refraction  of  a  wave  whose  normal  makes  an  angle 
<p  with  the  c  axis  of  a  uniaxial  crystal.     It  is  the  equation  of  an  ellipse  which  is  the 
inverse  of  that  of  equation  (8). 
From  Fig.  128  we  have 

x 

tan  «=     .  (10) 


Substituting  values  from  equations  (7),  we  have 

E2  sin  <f     Ez 


and,  further,  substituting  the  values  O  =  -  and  £  =  -,  we  have 


"  i     \ 

tan  «  =  --  tan  <p.  (12) 

This  is  an  equation  from  which  the  angle  <f>  may  be  obtained,  knowing  the  maximum 
and  minimum  indices  of  refraction  of  the  substance  and  the  angle  a. 

54.  Vibration  Directions  in  Uniaxial  Crystals.  —  Let  Fig.  130  represent  a 
principal  section  through  a  crystal  of  calcite,  MY  being  parallel  to  crystal- 
lographic  c.  A  ray  of  light  P,  entering  the  crystal  at  M,  will  be  broken  up 


74 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  54 


into  two  rays,  an  ordinary  ray  MO,  and  an  extraordinary  ray  ME.  At  the 
end  of  a  unit  of  time,  the  disturbance  of  the  ordinary  ray  will  have  reached 
the  point  O,  while  at  the  same  time  that  of  the  extraordinary  ray  will  have 
reached  E. 

We  have  seen  that  the  ordinary  ray  behaves  as  does  ordinary  light  in  an 
isotropic  medium,  consequently  its  vibrations  will  take  place  at  right  angles 
to  the  direction  of  propagation  and,  following  Fresnel,  perpendicular  to  a 
principal  section.  In  Fig.  130,  consequently,  these  vibrations  are  represented 
by  the  dots  between  M  and  N.  The  vibrations  of  the  extraordinary  ray 
take  place  at  an  angle  to  its  line  of  propagation  and  in  the  plane  of  crystallo- 


FIG.   130. — Vibration  directions  of  light  passing  through  an  anisotropic  medium. 

graphic  c,  consequently  they  lie  in  the  plane  of  the  paper  and  are  represented 
by  the  short  lines.  Whether  the  direction  of  vibration  is  at  right  angles 
to  the  ray  (ME)  itself,  or  to  the  normal  (MO)  to  the  wave  front,  is  unknown. 
Fresnel1  first  assumed  that  it  was  perpendicular  to  the  ray  and  therefore 
formed  an  angle  with  the  wave  front,  but  later2  he  decided  that  it  was  at 
right  angles  to  the  normal  and  thus  formed  an  angle  with  the  ray.  The 
latter  direction  seems  the  more  probable,  and  is  the  one  assumed  in  the 
electromagnetic  theory  of  light.3 

1  A.  Fresnel:  Memoire  sur  la  double  refraction.     Read,  Nov.  26,  1821.     Mem.  Acad- 
France,  VII  (1827),  45-176. 

Idem:  Oeuires  completes,  Paris,  1868,  II,  287. 

2  Idem:  Ibidem,  II,  339.     Read,  Jan.  23,  1882. 

3  J.  Clerk  Maxwell:  Electricity  and  magnetism,  Oxford,  1881,  II,  404. 

R.  T.  Glazebrook:  On  the  application  of  Sir  William  Thomson's  theory  of  a  contractile 
(Ether  to  double  refraction,  dispersion,  metallic  reflexion,  and  other  optical  problems.  Phil. 
Mag.,  XXVI  (1888),  521-540. 

G.  F.  Fitzgerald:  Electromagnetic  radiations.     Nature,  XLII,  1890,  172-175. 


ART.  55] 


ANISOTROPIC  MEDIA 


75 


55.  Ray  Surface  and  Wave  Surface  in  Uniaxial  Crystals. — If  we  compare 
equations  (i)  and  (80),  we  shall,  see  that  the  former  is  the  equation  of  an 
ellipse  while  the  latter  is  that  of  an  oval,  differing  slightly  in  form  from  the 
former  and  coinciding  only  along  the  diameters. 

The  surface  reached  by  all  rays  at  the  end  of  a  unit  of  time  is  known  as 
the  ray  surface,  that  reached  by  the  waves,  the  wave  surface. 

The  form  of  the  extraordinary  ray  surface,  as  we  shall  find,  is  an  ellipsoid, 
which,  in  uniaxial  crystals,  is  one  of  rotation,  oblate  for  negative  crystals, 

and  prolate  for  positive. 

2-2     yz 
The  equation1  of  the  curve  of  ray  fronts  ^+™  =  i,  is  that  of  an  ellipse.    By 

making  x  =  p,  combining  it  with  the  functional  equation  of  a  surface  of  revolution 
(pz  =  x*+yz),  and  changing  the  coordinates  so  that  the  Y  axis  extends  from  front  to 
back  and  the  Z  vertical, 


This  is  the  equation  of  the  ray  surface  of  a  uniaxial  crystal. 


FIG.   131. — Ray  surface  (solid  line),  wave  surface  (dotted  line),  ease  of  vibration  ellipsoid  (broken 
line),  and  Fresnel  ellipsoid  (dot  and  dash  line),  in  a  negative,  uniaxial  crystal  (O<E,  «>«). 

The  wave  surface  of  a  uniaxial  crystal  is  a  surface  of  rotation.  Fresnel 
considered  it  as  developed  by  a  system  of  plane  waves  starting  at  the  same 
time  from  the  center  of  a  crystal,  and  traveling  in  different  directions  along 
the  normals  (MN,  Fig.  131)  with  velocities  depending  upon  the  direction  of 
propagation.  At  the  end  of  any  instant  of  time  all  of  the  waves  will  be  tan- 
gent to  it.  The  position  of  the  point  N  of  the  dotted  curve,  which  represents 

!Eq.  i,  Art.  52,  supra. 


76 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  56 


the  wave  surface,  corresponds  to  the  point  R  of  the  curve  representing  the 
ray  surface.  At  the  points  where  x  or  y  =  O,  that  is,  on  the  axes,  the  ray 
and  normal  coincide,  consequently  the  two  surfaces  meet. 

The  equation  of  the  wave  fronts  represents  an  oval  of  the  form 


E2*2+OV=(*2+:y2)2- 


Eq.  8  Art.  53 


Combining  with  the  functional  equation  of  a  surface  of  revolution,  and  changing 
the  coordinates  so  that  the  F  axis  extends  from  front  to  back  and  the  Z  axis  is  ver- 
tical, we  have,  after  making  x=p, 

(*2+;y2+s2)2,  (14) 


the  equation  of  the  wave  surface  of  a  uniaxial  crystal. 

Along  each  of  the  three  axes  the  ray  and  wave  surfaces  coincide,  for  if  we  make, 
for  example,  y  and  z  equal  to  zero,  equations  (13)  and  (14)  alike  become  E2  =  x2. 

The  ray  and  wave  surfaces  of  the  ordinary  ray  coincide,  and  appear  in  section 
as  a  circle  (Fig.  131).  That  they  form  a  sphere  in  space  may  be  proved  by  making 
the  values  of  the  ordinary  and  extraordinary  rays  equal  in  equations  (13)  and  (14). 
The  former  becomes 

*2+r4-s2  =  £2orO2,  (15) 

and  the  latter 


(16) 


which  equals 

In  each  case,  the  equation  is  that  of  a  sphere. 


FIG.  132. — Propagation  of  light  in  a  uni- 
axial crystal,  forming  uniaxial  wave  and  ray 
surfaces. 


56.  Graphic  Development  of  Ray  and 
Wave  Surfaces  of  a  Uniaxial  Crystal.— 
We  may  now  develop  the  ray  and 
wave  surfaces  graphically.  Let  MC, 
Fig.  132,  be  the  c  axis  of  a  uniaxial 
negative  crystal;  all  vibrations  taking 
place  parallel  to  the  BMA  plane,  there- 
fore, will  be  equal.  Since  the  ease  of 
vibration  is  a  measure  of  the  rate  of 
propagation  or  velocity  of  a  ray,  we 
can  determine  the  position  of  any  ray 
front  or  wave  front  at  the  end  of  any 
instant  of  time. 


Let  O  =  velocity  of  the  ordinary  ray, 
E  =  velocity  of  the  extraordinary  ray, 
co  =  index  of  refraction  of  the  ordinary  ray, 
e    =  index  of  refraction  of  the  extraordinary  ray. 
We  have  already  determined1  that 

1  Art.  40,  supra. 


ART.  56]  AXISOTROPIC  MEDIA  77 

w  =  x,  o  =  --f  and£=I- 

fl  CO  € 

If  the  mineral  is  negative,  to > c,  and  O<E.  AM  =  MB  =  direction  of 
vibration  of  the  ordinary  ray  (index  =  co).  MC  =  direction  of  vibration  of 
the  extraordinary  ray  (index  =  e). 

If  a  disturbance  starts  at  M,  there  will  be  two  rays  of  light  passing  from 
M  to  A.  One,  the  ordinary  ray,  will  vibrate  parallel  to  MB.  Its  index  of 

refraction   =  co,  its   velocity  =  ->  and,  at  the  end  of  any  interval  of  time  (/), 

it  will  have  traveled  along  M  A l  a  distance  of  —  In  the  same  period  of  time, 
the  other  vibration,  the  extraordinary  ray,  vibrating  parallel  to  MC  with 
an  index  =  e,  will  travel  a  distance  of  •  The  vibrations  are  shown  in  the 

€ 

figure  by  the  short  vertical  lines  parallel  to  MC  and  along  MA.     Since 

/      / 
co>€,    -< 

co      e 

There  will  be,  also,  two  rays  traveling  along  MB;  the  ordinary  ray,  vibrat- 
ing parallel  to  MA  with  an  index  of  co  and  traveling  a  distance  of  ->  and  the 
extraordinary  ray,  vibrating  parallel  to  MC,  with  an  index  of  e,  and  traveling 

a   distance   of    -- 

e 

In  a  similar  manner  there  will  be  two  rays  propagated  along  MC,  one 
ray  with  an  index  of  co,  vibrating  parallel  to  MA,  and  traveling  a  distance  of 

;  and  another  with  an  index  of  co,  vibrating  parallel  to  MB,  also  traveling 

a  distance  of  —    That  is,  in  this  direction  both  rays  will  reach  the  eye  at  the 

same  time,  a  fact  which  we  had  already  ascertained  by  our  examination  of 
the  calcite  rhombohedrons. 

So  far  we  have  considered  the  two  rays  vibrating  along  each  of  the  three 
coordinate  axes.  As  we  have  already  seen,2  along  these  axes  the  light  ray 
and  the  normal  to  the  wave  front  coincide  since  the  tangents  to  an  ellipse 
at  the  end  of  the  axes  lie  at  right  angles  to  these  axes;  the  tangents  represent- 
ing the  directions  of  vibration,  and  the  axes,  the  normals  to  the  wave  front  and 
also  the  lines  of  propagation  of  the  rays. 

Consider  now  the  plane  A  MB.  Since  the  crystal  under  examination  is 
uniaxial,  all  vibrations  in  this  plane  are  equal,  and  any  ray,  as  MX  and  MX', 

will  reach  distances  of  —  and  —  at  the  end  of  the  time  /,  whereby  the  ray 

un  e 

1  In  Fig.  130,   to  avoid 'confusion,  the  vibrations  are  shown,  not  bisected  by  the  line 
MA,  etc.,  but  on  one  side  only. 

2  Art.  55,  supra. 


78  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  56 

fronts  of  all  of  the  ordinary  rays  in  the  plane  AMB  will  lie  along  AXB,  a 
circle,  and  those  of  the  extraordinary  ray  along  A'X'B',  also  a  circle.  Now 
a  tangent  to  a  circle  is  perpendicular  to  a  radius,  consequently  the  vibrations 
of  all  rays,  both  extraordinary  and  ordinary,  act  at  right  angles  to  the  direc- 
tion of  transmission  of  the  ray.  Since  this  is  the  case,  the  normal  to  the  wave 
front  coincides  in  direction  with  the  direction  of  propagation  of  the  ray, 
whereby  the  curves  of  the  ray  and  wave  fronts  are  the  same.1 

Let  us  see  what  this  means.  If  the  normal  to  the  wave  front  of  the  ex- 
traordinary ray  coincides  with  the  direction  of  propagation  of  the  ordinary 
ray,  the  two  rays  must  be  propagated  along  the  same  line  so  that,  if  we  were 
to  look  through  a  uniaxial  crystal  along  any  line  in  the  plane  AMB,  that  is, 
along  any  line  perpendicular  to  the  c  axis,  we  should  see  but  a  single  image. 
That  such  is  actually  the  case  we  have  already  seen  in  the  case  of  calcite. 
That  the  two  rays  do  not  reach  the  eye  at  the  same  time,  however,  and  thus 
differ  from  the  rays  along  crystallographic  c,  we  can  determine  by  a  measure- 
ment of  the  retardation — a  measurement,  as  we  shall  find  later,  which  can 
be  made  under  the  microscope  by  means  of  polarized  light. 

In  no  other  direction,  however,  do  the  two  curves  coincide.  For  example, 
a  ray  from  M ,  traveling  in  the  plane  CM  A ,  will  be  broken  up  into  two  rays, 
the  ordinary  ray  MK  with  vibrations  parallel  to  MB  and  at  right  angles  to 
the  direction  of  propagation,  and  the  extraordinary  ray  MR  with  vibrations 
parallel  to  the  tangent  to  its  ray  front,  the  ellipse  CRA',  at  R. 

The  distance  traveled  in  any  direction  in  this  plane  by  the  ordinary  ray 

in  the  time  /  will  be  — .     Since  the  ray  and  the  normal  to  the  tangent  lie 

along  the  same  line,  the  ray  front  and  wave  front  coincide.  The  extraordi- 
nary ray,  however,  travels  a  distance  of  /  times  the  value  of  equation  3, 
Art.  52,  and  its  surface  is  the  ellipse  given  by  this  equation.  Its  major  and 
minor  axes  are  shown  by  MA'  and  MC,  Fig.  132.  The  wave  front  (MN) 

travels  a  distance  equal  to  -;  its  curve  is  given  by  equation  (8)  and  is  shown 

graphically  by  the  broken  line  CNA',  Fig.  132. 

In  the  plane  BMC,  both  ray  front  and  wave  front  of  the  rays  which  have 
their  vibrations  at  right  angles  to  that  plane  (the  ordinary  rays)  will  lie  on 
the  circle  CB.  The  rays  whose  vibrations  lie  within  the  BMC  plane  (the 
extraordinary  rays)  have  for  their  ray  front  the  curve  shown  by  the  solid 
line  between  C  and  B',  an  ellipse,  while  their  wave  front  is  shown  by  the 
dotted  curve  between  the  same  points,  an  oval. 

From  this  construction  we  can  see  that  the  two  rays,  the  ordinary  and 
the  extraordinary,  may  be  considered  as  forming  two  double  surfaces.  The 
ray  surface  of  the  ordinary  ray  is  a  sphere  (whose  equation  is  given  by  equation 
15)  of  which  CABC  (Fig.  132)  is  the  part  appearing  in  the  upper  front  right- 

1  Art.  55,  supra. 


ART.  57] 


ANISOTROPIC  MEDIA 


79 


hand  octant.  This  form  of  surface  was  to  have  been  expected,  since  the  ordi- 
nary ray  acts  like  a  ray  of  light  in  an  isotropic  substance,  in  which  the 
vibrations  are  equal  in  all  directions.  The  ray  surface  of  the  extraordinary 
ra\  has  for  its  section  in  AMB,  a  circle,  while  the  sections  in  CMB  and  CM  A 
are  similar  ellipses.  In  any  other  plane,  as  CMX,  the  vertical  section  of  the 
extraordinary  ray  is  also  a  similar  ellipse,  consequently  the  ray  surface  is  an 
ellipsoid  of  rotation  (proved  by  equation  13).  The  wave  surface  of  the  ex- 
traordinary ray  has  for  its  section  in  AM B  the  same  circle  as  the  ray  surface, 
therefore  the  two  rotation  surfaces  coincide  along  this  line.  In  the  planes 
CMB  or  CM  A ,  however,  or  in  any  intermediate  plane,  the  vertical  section  of 
the  wave  front  of  the  extraordinary  ray  is  an  oval,  consequently  the  wave 
surface  is  a  spheroid  of  rotation  (proved  by  equation  14). 


FIG.   133.  FIG.    134. 

FIGS.   133  TO  134. — Wave  surfaces  of  positive  and  negative  uniaxial  crystals. 

extraordinary  ray. 


O  =  ordinary  ray,  £  = 


In  the  case  considered,  the  value  of  a>  was  taken  as  greater  than  that 
of  e,  and  in  the  surfaces  developed  the  sphere  lies  within  the  ellipsoid  or 
spheroid,  which  is  oblate  (Fig.  134).  In  positive  crystals,  with  w<e,  a 
prolate  ellipsoid  or  spheroid  lies  within  the  sphere  (Fig.  133). 

57.  Curve  of  Ease  of  Vibration  (Fresnel's  Curve  of  Elasticities).  —  The  distance 
which  a  ray  or  a  wave  travels  depends  upon  the  ease  of  vibration  in  a  direction  at 
right  angles  to  the  normal,  that  is,  the  greater  the  ease  of  vibration  in  a  certain 
direction,  the  greater  the  velocity  of  the  wave  advancing  at  right  angles  to  this 
direction.  Thus  in  Fig.  131,  a  wave  advancing  from  M  to  N  and  a  ray  from  M  to  R 
will  have  their  vibrations  in  the  direction  of  MV. 


Using  this  value  for  the  angle  in  the  equation  to  the  curve  of  wave  fronts  (8), 
we  have 

Ez  sin2  (9o°-*>)+O2  cos2  (go°-<p)=w2, 
or  £2  cos2  H-02  sin2  ^  =  w2. 


But 


sin  <f 


which,  substituted  in  (17),  gives 


w  ' 


, 


80  MANUAL  OF  PETROGRAPHIC  METHODS  [Asx.  58 


or  E^i2+02*i2=(*i2+3'i2)2.  (18) 

This  is  the  equation  of  an  oval  exactly  like  the  oval  of  wave  fronts  but  with  its 
long  axis  at  right  angles  to  the  long  axis  of  the  latter.  It  is  shown  in  Fig.  131  by 
the  broken  line.  This  curve  gives  both  the  ease  and  the  direction  of  vibration  anywhere 
within  a  crystal. 

Combining  (18)  with  the  functional  equation  of  a  surface  of  revolution,  and  using 
z  for  the  vertical  axis,  we  obtain  a  surface  like  that  given  in  (14)  except  that  it  has 
its  major  and  minor  axes  at  right  angles  to  the  major  and  minor  axes  of  the  latter. 
Its  axis  of  rotation  is  the  axis  extending  from  left  to  right.  Since  p  =  x 


2)'.  (19) 

It  is  the  equation  of  a  spheroid  and  is  the  surface  of  ease  of  "vibration. 

58.  Fresnel's  Ellipsoid.  —  The  Fresnel  ellipsoid  is  a  simple  ellipsoid  of  one 
surface  which  has  for  its  three  axes  the  maximum,  minimum,.  and  mean  ease 
of  vibration.  In  uniaxial  crystals  the  mean  ease  coincides  with  one  of  the 
other  values,  and  the  figure  is  an  ellipsoid  of  rotation.  Along  the  axes,  this 
ellipsoid  coincides  with  the  single  surface  ease  of  vibration  spheroid  (equation 
19).  It  bears  the  same  relation  to  the  ray  surface  as  does  the  ease  of  vibra- 
tion spheroid  to  the  wave  surface,  and  is  shown  in  Fig.  131  by  the  solid  line 
YFY'.  The  line  MF  does  not  represent  the  vibration  direction  of  the  ray 
MR. 

The  equation  of  the  Fresnel  ellipsoid  may  be  obtained  as  follows: 
The  inverse  of  the  equation  of  the  ray  front  curve  (i)  is 

*»     y* 

O2~TE2 

which,   changed  to  coordinates  with  the  z  axis  vertical,  and  combined  with  the 
functional  equation  of  a  surface  of  revolution,  becomes 


//  is  the  equation  of  the  Fresnel  ellipsoid. 

59.  The  Optical  Indicatrix.  —  Another  method  of  representing  the  optical 
characters  of  crystals  is  by  means  of  a  figure  based  upon  its  indices  of  refrac- 
tion. This  surface  of  reference  was  called  the  ellipsoid  of  indices  by  Mac- 
Cullagh,  ellipsoide  des  elasticities  by  Fresnel,  indexellipsoid  by  Liebisch, 
and  optical  indicatrix,  or  simply  indicatrix,  by  Fletcher.1 

In  Fig.  135  let  Mr  represent  the  direction  of  propagation  and  the  velocity 
of  a  ray  of  light. 

1  L.  Fletcher:  The  optical  indicatrix  and  the  transmission  of  light  in  crystals.  London, 
1892,  20. 


ART.  59] 


ANISOTROPIC  MEDIA 


81 


MC  and  MA,  and  Mr  and  MR  are  two  pairs  of  conjugate  radii. 

Draw  RN  perpendicular  to  Mr. 

From  the  property  of  an  ellipse,  we  have:  The  area  of  a  parallelogram 
circumscribing  an  ellipse  in  which  the  sides  are  tangent  to  the  ellipse  at  the 
vertices  of  a  pair  of  conjugate  diameters,  is  constant  and  equal  to  the  rect- 
angle constructed  on  the  axes. 

Therefore  MA'MC  =  MrRN  =  k,  a  constant; 

MA-MC       k          _       k 
hence  Mr  = 


That  is,  RN  and  Mr  are  inversely  proportional  to  each  other,  no  matter 
what  the  direction  of  the  ray  Mr. 

Now 


i 
»=-> 

v 


FIG.   135. — Geometrical  relations  in  an  ellipse.          FIG.   136. — Relation  between  indicatrix  and  ray 

surface. 

therefore,  if  Mr  represents  the  distance  traveled  by  the  light  in  the  time  k} 

k 


But 


"  —   njr     ' 

Mr 

n  =  c  or  co,  and  Mr  =  E  or  O  = 


e  or 


If  the  crystal  under  consideration  is  uniaxial  and  negative,  E>O  and 
e  <  co,  consequently 

Mr  (the  major  axis)  =  E,  and  RN  =  O; 


also 


MA=E=-     and      MC=0=k< 

€  OJ 


That  is,  if  MA  (or  Mr)  represents  the  velocity  of  a  ray  of  light,  the  normal 
from  the  vertex  of  its  conjugate  CM  (or  RN)  will  represent  its  index  of 
refraction  multiplied  by  a  constant. 

The  indicatrix,  then,  is  an  ellipsoid  of  rotation  whose  major  radius  is 
equal  to  its  maximum  index  of  refraction,  and  whose  minor  axis  is  equal  to 
its  minimum  index  of  refraction. 

For  example  (Fig.  136),  if  u=i.8  =  MA,  and  e=i.2=MC,  at  the  end  of  a 
unit  of  time 

6 


82 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  60 


o        E.., 

1.8'  1.2 

as  is  shown  by  the  inner  ellipse.     In  the  ordinary  ray  o>  =  e  =  i .  2  and  O  —  E 
— ,  as  shown  by  the  inner  circle. 


i .  2 


The  indicatrix  for  a  positive  crystal  will  be  prolate  instead  of  oblate. 

Analytically  the  equation  of  the  index  surface  may  be  obtained  as  follows: 
The  values  of  the  major  and  minor  axes  of  the  curve  of  indices  in  a  negative 

crystal  are  co  for  the  major  and  e  for  the  minor  axis.     Substituting  these  values  in 

the  equation  of  an  ellipse  we  have 


or 


-2+2  = 

Or        t* 


which,  combined  with  the  functional  equation  of  a  surface  of  rotation  gives,  after 
changing  the  vertical  axis  to  z, 


This  is  the  equation  of  the  optical  indicatrix  of  a  negative  uniaxial  crystal. 

60.  Huygens'  Construction  for  Double 
Refraction  in  Uniaxial  Crystals.  —  Refrac- 
tion of  light  in  an  isotropic  medium  may  be 
shown  in  another  manner: 

In  Fig.  137  let  /  and  /'  be  two  parallel 
rays  of  light  meeting  the  surface  X'X  at 
M  and  B.  When  the  ray  I  is  at  M,  /' 
is  at  N.  If  /  were  free  to  go  on  without 
change  of  velocity  it  would  reach  the  point 
R  when  I'  reached  B. 
Draw  a  circle  with  M  as  a  center  and  NB  as  a  radius.  The  wave  front 
RB  is  evidently  tangent  to  it  at  R.  Now  draw,  with  M  as  a  center,  a  circle 
having  a  radius  of  MD,  equal  to  the  distance  traveled  by  the  ray  in  the  denser 
medium  in  the  same  length  of  time  that  is  required  to  travel  from  N  to  B  in 
air.  The  line  DB,  drawn  through  the  point  B  and  tangent  to  the  circle  at  D, 
will  evidently  give  the  wave  front  of  the  ray  in  the  denser  medium,  and  the 
radius  MD,  perpendicular  to  DB,  the  direction  of  propagation  of  the  ray. 

In  an  anisotropic  uniaxial  crystal,  as  we  have  seen,  light  is  separated  into 
two  rays.     We  may  have  several  cases: 


B — x 


FIG.   137. — Refraction  of  light  in  an  iso- 
tropic medium. 


I.  THE  OPTIC  Axis  is  PERPENDICULAR  TO  THE  PLANE  OF  INCIDENCE 

a.  The   crystal  is  negative.     Let  Fig.  138  represent  a  crystal  of  calcite 
with  its  c  axis  perpendicular  to  the  plane  of  the  paper.     In  this  case  the 


ART.  60] 


ANISOTROPIC  MEDIA 


83 


sections  through  the  ray  surfaces  will  appear  as  two  circles,  as  shown  in  the 
drawing,  the  radii  being  proportional  to  the  velocities  of  the  ordinary  and 
extraordinary  rays,  that  is,  inversely  proportional  to  their  indices.  The 
wave  fronts  of  the  two  rays,  7  and  /',  after  passing  into  the  second  medium, 
will  lie  on  lines  passing  through  B  and  the  point  of  tangency  of  their  respec- 


FIG.   138. 


„  FIG.   139. 

FIG.  138. — Refraction  of  light  in  a  negative,  uniaxial  crystal  with  the  plane  of  incidence  perpen- 
dicular to  the  optic  axis. 

FIG.  139. — Refraction  of  light  in  a  negative,  uniaxial  crystal  with  the  plane  of  incidence  perpen- 
dicular to  the  optic  axis  and  with  the  incident  ray  perpendicular  to  the  reflecting  surface. 


tive  circles.  Since  in  calcite  E>O  and  e<a>,  the  shorter  radius  MO 
(0  being  the  point  of  tangency)  will  be  the  direction  of  propagation  of  the 
ordinary  ray,  and  the  longer  radius  ME  will  be  the  direction  of  propagation 
of  the  extraordinary  ray. 

If  the  incident  ray  is  normal  to  the  reflecting  surface  (Fig.  139),  the  tan- 
gents to  the  three  circles  are  parallel  to  each  other  and  at  right  angles  to  the 
ray. 


FIG.   140.  FIG.  141. 

FIG.  140. — Refraction  of  light  in  a  positive,  uniaxial  crystal  with  the  plane  of  incidence  perpen- 
dicular to  the  optic  axis. 

FIG.  141. — Refraction  of  light  in  a  positive,  uniaxial  crystal  with  the  plane  of  incidence  perpen- 
dicular to  the  optic  axis  and  with  the  incident  ray  perpendicular  to  the  reflecting  surface. 

From  these  two  cases  we  see  that  the  ordinary  ray  is  more  deflected  than 
the  extraordinary  ray,  in  negative  crystals,  when  the  plane  of  incidence  is 
perpendicular  to  the  optic  axis.  An  exception  occurs  in  the  case  of  normal 
incidence  where  neither  ray  is  deflected,  although  the  extraordinary  ray 
travels  faster  than  the  ordinary. 


84 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  60 


b.  The  crystal  is  positive.  In  positive  crystals,  with  O>E  and  a><e  the 
length  of  the  radius  of  the  ordinary  ray  surface  is  greater  than  that  of  the 
extraordinary  (Fig.  140),  consequently,  by  the  same  construction  as  above, 
it  is  seen  that  the  extraordinary  ray  is  more  deflected  than  the  ordinary  ray. 
When  the  incident  ray  is  perpendicular  to  the  reflecting  surface  there  is  no 


FIG.  142.  FIG.   143. 

FIG.  142. — Refraction  of  light  in  a  negative,  uniaxial  crystal  with  the  optic  axis  parallel  to  both 
the  plane  of  incidence  and  the  reflecting  surface. 

FIG.  143. — Refraction  of  light  in  a  negative,  uniaxial  crystal  orientated  as  in  the  preceding  figure 
but  with  the  incident  ray  perpendicular  to  the  reflecting  surface. 

deflection  (Fig.  141),  but  the  ordinary  ray  reaches  the  eye  sooner  than  the 
extraordinary. 

II.  THE  OPTIC  Axis  LIES  IN  THE  PLANE  OF  INCIDENCE  AND  is  PARALLEL 
TO  THE  REFLECTING  SURFACE 

a.  Negative  Crystals. — In  a  negative  crystal,  when  the  optic  axis  lies  in 
the  plane  of  incidence  and  is  parallel  to  the  reflecting  surface,  the  extraordi- 


FIG.   144.  FIG.   145. 

FIG.  144. — Refraction  of  light  in  a  positive,  uniaxial  crystal  with  the  optic  axis  parallel  to  both 
the  plane  of  incidence  and  the  reflecting  surface. 

FlG.  145. — Refraction  of  light  in  a  positive,  uniaxial  crystal  orientated  as  in  the  preceding  figure 
but  with  the  incident  ray  perpendicular  to  the^eflecting  surface. 

nary  ray  is  deflected  more  than  the  ordinary,  as  may  be  seen  from  Fig.  142. 
When  the  incident  ray  is  normal  to  the  reflecting  surface  neither  ray  is 
deflected,  but  the  extraordinary  ray  reaches  the  eye  before  the  ordinary 
(Fig.  143). 


ART.  00] 


ANISOTROPIC  MEDIA 


85 


b.  Positive  Crystals. — In  a  positive  crystal  the  ordinary  ray  is  more  de- 
flected than  the  extraordinary  (Fig.  144).  When  the  incident  ray  is  normal 
to  the  reflecting  surface  neither  ray  is  deflected,  but  the  ordinary  ray  reaches 
the  eye  before  the  extraordinary  (Fig.  145). 

III.  THE  OPTIC  Axis  LIES  IN  THE  PLANE  OF  INCIDENCE  AND  is  PERPEN- 
DICULAR TO  THE  REFLECTING  SURFACE 

a.  Negative  Crystals. — In  a  negative  crystal,  when  the  optic  axis  lies  in 
the  plane  of  incidence  but  is  perpendicular  to  the  reflecting  surface,  the  ordi- 


FIG.   146. 


FIG.   147. 


FIG.   146. — Negative,  uniaxial  crystal,  optic  axis  vertical. 

FIG.   147. — Negative,  uniaxial  crystal,  optic  axis  vertical,  incidence  normal. 

nary  ray  is  deflected  more  than  the  extraordinary  (Fig.  146).  When  the 
incident  ray  is  normal  to  the  reflecting  surface,  there  is  neither  deflection 
nor  retardation,  both  rays  reaching  the  eye  at  the  same  time  (Fig.  147). 

b.  Positive  Crystals. — In  a  positive  crystal  with  its  optic  axis  perpen- 
dicular to  the  reflecting  surface,  the  extraordinary  ray  is  deflected  more  than 


FIG.   148. 


FIG.   149. 


FIG.   148.  —  Positive,  uniaxial  crystal,  optic  axis  vertical. 

FIG.    149.  —  Positive,  uniaxial  crystal,  optic   axis   vertical,  incident  ray  normal  to  reflecting  plane. 

the  ordinary  (Fig.  148).  When  the  incident  ray  is  normal  to  the  reflecting 
surface,  there  is  neither  deflection  nor  retardation,  both  rays  reaching  the 
eye  at  the  same  time  (Fig.  149). 


86 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  60 


IV.  THE  OPTIC  Axis  LIES  IN  THE  PLANE  or  INCIDENCE  BUT  FORMS 
AN  ANGLE  WITH  THE  REFLECTING  SURFACE 

When  the  optic  axis  is  inclined  to  the  reflecting  surface,  the  relative 
refraction  of  the  ordinary  and  the  extraordinary  rays  varies  with  the  inclina- 
tion of  the  axis.  This  may  be  seen  by  an  inspection  of  Figs.  150  and  151, 
which  show  the  refraction  of  various  rays  in  a  negative  uniaxial  crystal. 

A  comparison  of  the  variations  found  when  the  optic  axis  is  parallel, 
perpendicular,  or  inclined  to  the  reflecting  surface  is  shown  in  Fig.  152. 


FIG.  150. — Negative,  uniaxial  crystal,  optic  axis  inclined  and  lying  in  the  plane  of  incidence  which 
is  perpendicular  to  the  reflecting  surface.  The  optic  axis  lies  beyond  the  critical  angle,  a  and  a', 
complements  of  the  critical  angle  for  the  ordinary  ray ;  /3  and  0',  complements  of  the  critical  angle  for 
the  extraordinary  ray. 

Points  from  A  to  G  represent  the  rays  A  to  G  (Figs.  150  and  151),  and 
horizontal  distances,  the  deflection  to  the  right  or  left  from  the  direction 
of  the  incident  ray  measured  on  any  circle.  Deflection  in  the  direction  of 
the  movement  of  the  hands  of  a  watch  (negative  deflection)  is  shown  to 
the  left  of  the  vertical  line,  while  deflection  in  the  opposite  direction  (positive 
deflection)  is  shown  to  the  right  (Fig.  152).  The  vertical  line  represents 
the  direction  of  the  incident  ray. 

A  comparison  of  Figs.  150  and  151  with  152  will  show  the  following 
facts: 

i.  THE  ORDINARY  RAY. — No  matter  how  the  crystal  is  oriented,  the 
deflection  of  the  ordinary  ray  (curve  OO,  Fig.  152)  remains  the  same.  Light 


ART.  60] 


ANISOTROPIC  MEDIA 


87 


entering  between  A  and  G  (Figs.  150-151)  will  be  refracted  within  a  space 
equal  to  double  the  critical  angle,  that  is,  a  space  measured  on  either  side  of 
the  normal  (Dd)  by  the  critical  angle.  Rays  between  o°  and  90°  (A  to  D) 
will  be  deflected  in  the  negative  direction,  the  90°  ray  (D)  will  pass  straight 
through,  and  rays  between  90°  and  180°  (D  to  G)  will  be  deflected  in  the 
positive  direction.  The  critical  angle  for  the  ordinary  is  the  same  on  either 
side  of  the  normal  (AO=GO,  Fig.  152)  and  the  curve  is  a  straight  line,  crossing 
the  vertical  at  D  (90°)  where  there  is  no  deflection. 


c-. 


FIG.  151. — Negative,  urraxial  crystal,  optic  axis  inclined  and  lying  in  the  plane  of  incidence  which 
is  perpendicular  to  the  reflecting  surface.  The  optic  axis  falls  within  the  critical  angle,  a,  a',  (I,  and 
/3'  represent  the  same  angles  as  in  the  preceding  figure. 

II.  THE  EXTRAORDINARY  RAY. — i.  The  optic  axis  lies  in  the  plane  of  in- 
cidence and  is  parallel  to  the  reflecting  surface  (Fig.  142  and  line  £-142,  Fig. 
152).  The  extraordinary  ray  is  symmetrically  deflected  when  the  optic  axis 
is  parallel  to  the  reflecting  surface.  The  maximum  deflection  on  either  side 
is  at  the  critical  angle,  all  other  rays  being  deflected  between  these  two 
points.  For  rays  entering  between  o°  and  90°  (^4  to  D)  the  deflection  is 
negative,  the  90°  ray  passes  straight  through,  and  between  90°  and  180°  the 
rays  are  deflected  in  a  positive  direction.  In  every  case  the  extraordinary 
ray  is  more  deflected  than  the  ordinary  ray,  as  we  have  already  seen  (Fig. 
142). 

2.  The  optic  axis  lies  in  the  plane  of  incidence  and  is  perpendicular  to  the 
reflecting  surface  (Fig.  146  and  line  £-146,  Fig.  152).  When  the  optic  axis  is 
perpendicular  to  the  reflecting  surface,  the  light  also  is  symmetrically  de- 
flected. The  maximum  value  is  at  the  critical  angle.  The  90°  ray  is  not 


88 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  60 


deflected.  Between  o°  and  90°  the  deflection  is  negative,  between  90°  and 
1 80°,  positive.  In  every  case  the  extraordinary  ray  is  less  deflected  than  the 
ordinary  (cf.  Fig.  146). 

3.  The  optic  axis  lies  in  the  plane  of  incidence  but  forms  an  angle  with  the 
reflecting  surface.  When  the  optic  axis  is  inclined  to  the  normal  at  more 
than  the  critical  angle  (Figs.  150  and  lines  £-150,  Enw,  and  Ene,  Fig.  152), 
the  extraordinary  ray  is  more  or  less  deflected  than  the  ordinary,  depending 
upon  the  position  of  the  optic  axis. 


FIG.  152. — Diagram  showing  deflection  to  the  right  or  left  of  the  incident  rays  of  Figs.  142,  146,  150 
and  151.  The  ordinary  ray  is  uniformly  deflected,  irrespective  of  the  position  of  the  crystal.  Ene 
and  Enw  show  deflections  in  the  crystal  when  it  is  placed  with  its  optic  axis  respectively  at  45°  to  the 
normal  to  the  reflecting  surface  in  the  upper  right  quadrant  and  in  the  upper  left  quadrant.  Deflection 
is  considered  negative  when  in  a  counter-clock-wise  direction. 


When  the  optic  axis  is  inclined  to  the  normal  at  less  than  the  critical 
angle  (Fig.  151  and  line  £-151,  Fig.  152)  there  will  be  some  point  where 
both  the  ordinary  and  the  extraordinary  rays  are  equally  deflected.  This 
position  is  reached  when  the  refracted  ray  lies  along  the  optic  axis.  In  this 
position  there  is  no  double  refraction,  although  the  ray  is  deflected  except 
when  the  incident  ray  is  at  90°.  The  position  of  no  double  refraction  is 
shown  in  Fig.  152  where  the  O  and  £-151  curves  cross;  the  relative  amounts 
of  deflection  are  indicated  by  the  distances  to  the  left  (or  right)  of  the 
vertical. 

It  is  to  be  noted  that  wherever  an  "E"  curve  crosses  the  vertical  line, 


ART.  61]  ANISOTROPIC  MEDIA  89 

Fig.  152,  there  is  no  deflection  of  the  extraordinary  ray,  but  it  passes  straight 
through  the  crystal. 

Besides  the  deflection  curves  given  above,  many  others  may  be  con- 
structed for  positions  of  the  optic  axis  when  it  does  not  lie  in  the  plane  of 
incidence. 

For  positive  crystals  the  phenomena  above  described  are  reversed,  since 
in  such  crystals  the  velocity  of  the  ordinary  ray  is  greater  than  that  of  the 
extraordinary. 

61.  Summary  of  the  Optical  Properties  of  Uniaxial  Crystals. — From  our 
examination  of  the  effect  of  uniaxial  crystals  upon  the  passage  of  light 
through  them,  we  have  derived  the  following  data: 

1.  There  are  certain  substances  which  differ  from  isotropic  media  in 
that  the  light,  in  passing  through  them,  is  broken  up  into  two  rays  vibrating 
at  right  angles  to  each  other.     Such  substances  are  called  anisotropic  (Arts. 
47  and  48). 

2.  Of  the  two  rays  of  light  resulting  from  the  double  refraction  of  a  ray 
passing  through  a  uniaxial  crystal,  one  vibrates  with  equal  ease  in  every 
direction.     This  is  called  the  ordinary  ray  and  it  is  represented  by  the 
letter  O.     Its  ray  surface  is  a  sphere  and  its  index  of  refraction  is  denoted 
by  to. 

3.  The  second  ray  is  called  the  extraordinary  ray.     It  is  indicated  by  the 
letter  E  and  its  index  of  refraction  by  e.     It  passes  through  the  crystal  with 
different  velocities  in  different  directions,  agreeing  only  in  one  direction 
with  the  ordinary  ray.     In  this  direction,  consequently,  there  will  be  no 
double  refraction,  since  the  values  of  co  and  e  are  equal;  in  other  words, 
this  is  an  axis  of  isotropy.     In  uniaxial  crystals  this  axis  always  coincides 
with  crystallographic  c. 

4.  Crystals  belonging  to  the  tetragonal  and  hexagonal  systems  are  uni- 
axial.    In  them  the  axis  of  greatest  or  least  ease  of  vibration  coincides  with 
crystallographic  c,  the  optic  axis.     All  rays  forming  the  same  angle  with  this 
axis  have  equal  velocities,  consequently  any  section  of  the  ray  surface  at 
right  angles  to  this  axis  is  a  circle,  while  every  other  section  is  an  ellipse. 

5.  The  ray  surface  consists  of  two  sheets  or  shells  representing  the  veloci- 
ties of  the  ordinary  and  the  extraordinary  rays.     The  former  is  a  sphere,  the 
latter  an  ellipsoid  of  rotation  (Art.  55). 

6.  The  wave  surface  consists  of  two  sheets  or  shells  representing  the 
velocity  of  transmission  of  the  waves  produced  by  the  ordinary  and  the  ex- 
traordinary rays.     The  direction  of  propagation  of  a  wave  is  represented 
by  the  normal  to  a  tangent  to  the  ray  front,  while  the  direction  of  propa- 
gation of  a  ray  is  represented  by  a  radius  (Art.  53).     These  two  directions 
coincide  for  the  ordinary  ray,  since  the  ray  surface  is  a  sphere  and  the 
normal   to   the   tangent   at   the   surface  is  the  radius.     The  directions  of 


90  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  61 

propagation  being  the  same,  the  wave  and  ray  surfaces  likewise  coincide. 
The  direction  of  the  normal  in  the  ellipsoid  of  the  ray  surface  is  not 
the  same  as  the  radius  at  the  point  of  tangency,  consequently  the  extra- 
ordinary ray  surface  is  not  the  same  as  its  wave  surface.  The  former  is  an 
ellipsoid  of  rotation,  the  latter  a  spheroid  of  rotation  differing  slightly  from 
an  ellipsoid  (Art.  55). 

7.  There  are  two  kinds  of  uniaxial  crystals.     In  one  the  extraordinary 
ray  has  its  greatest  velocity  along  the  principal  axis;  in  the  other  this  is  the 
direction  of  least  velocity.     The  form  of  the  ray  surfaces  are  as  shown  in 
Figs.  133  and  134,  depending  upon  whether  the  velocity  of  0>-E  (6o<e),  or 
O<E  (a)>e).     The  former  are  called  positive  (+)  crystals,  the  latter  nega- 
tive (-)  (Art.  51). 

8.  A  principal  optic  section  is  one  containing  the  axes  of  greatest  and  least 
ease  of  vibration.     In  uniaxial  crystals  any  section  parallel  to  crystallo- 
graphic  axis  c  is  a  principal  section  (Art.  50). 


CHAPTER  VI 

ANISOTROPIC  MEDIA  (Continued) 
OPTICALLY  BIAXIAL  CRYSTALS 

62.  Vibration  Axes. — We  have  seen,  in  uniaxial  crystals,  that  light  vibrates 
in  one  plane  with  equal  ease  in  every  direction,  and  that  at  right  angles  to 
this  direction  is  the  position  of  maximum  or  minimum  ease  of  vibration. 
With  a  system  of  three  coordinates  we  would  have,  then,  x  =  y^z,  where 
x  and  y  are  the  horizontal  components  and  z  the  vertical. 

There  is  another  class  of  crystals  in  which  the  ease  of  vibration,  and 
consequently  the  indices  of  refraction,  differ  in  different  directions,  therefore 
x^y^z.  In  a  given  direction  in  a  crystal  this  ease  of  vibration  is  constant, 
and  the  three  chief  vibration  axes,  always  at  right  angles  to  each  other 
(Art.  47),  are  indicated  by  the  German  letters  0,  B,  and  C;1  ft  being  con- 
sidered the  direction  of  greatest  ease,  b  the  direction  of  intermediate  ease, 
and  C  the  direction  of  least  ease  of  vibration  (o>fi>c).  The  velocities 
of  light  corresponding  to  these  axes  vary  inversely  as  their  respective  indices 
-of  refraction,  as  has  been  shown  above  (Art.  40),  and  since  a  ray  is  propagated 
in  a  direction  at  right  angles  to  the  direction  of  its  vibrations,  along  these  axes 
it  will  advance  fastest  when  it  is  vibrating  parallel  to  a  and  slowest  when 
parallel  to  c.  The  planes,  always  at  right  angles  to  each  other,  in  which 
these  three  principal  vibration  axes  intersect,  are  called  the  principal  optic 
sections  of  the  biaxial  crystal. 

The  index  of  refraction  of  the  ray  with  vibrations  parallel  to  a,  and  ad- 
vancing at  right  angles  to  it,  is  represented,  in  biaxial  crystals,  by  a.  Taking  the 

velocity  in  vacuum  as  unity,  its  velocity  is  equal  to  — .    The  index  of  re- 

1  X,  Y,  and  Z  were  substituted  for  0,  B,  and  t  by  Iddings,  but  this  causes  confusion 
when  writing  equations  in  which  these  letters  are  used  also  for  coordinate  axes,  as  in  Art. 
288.  Wright  (The  index  ellipsoid  [optical  indicatrix]  in  petrographic  microscopic  work,  Amer. 
Jour.  Sci.,  XXXV  (1913),  133-138)  suggests  abandoning  the  "elasticity  ellipsoid"  and  the 
symbols  for  the  "axes  of  elasticity"  in  the  explanation  of  the  phenomena  of  light  in  crys- 
stals.  He  would  use  instead  only  the  indicatrix  and  the  symbols  for  the  refractive  indices, 
regarding  the  use  of  other  symbols  as  bewildering  to  the  student.  The  writer's  experi- 
ence has  been  that  students  can  grasp,  much  more  readily,  the  idea  of  an  ease  (or  diffi- 
culty) of  vibration  in  a  certain  direction  in  a  crystal,  and  a  corresponding  rate  of  movement 
at  right  angles  to  it,  than  they  can  the  inverse  relation  of  the  refractive  indices.  The 
writer  long  ago  abandoned  the  terms  "axes  of  elasticity"  and  substituted  for  them  "ease 
of  vibration  axes."  In  his  Determination  of  rock-forming  minerals,  1908  (pages  5,  6,  7,  9, 
n,  12, 13,  19,  21,  22,  24,  etc.,  etc.,  except  page  8  where  the  former  term  was  inserted  by  an 
oversight)  he  invariably  used  the  latter. 

91 


92  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  63 

fraction  of  the  ray  with  vibrations  parallel  to  b  and  advancing  at  right  angles 
to  it,  is  represented  by  /?.  Its  velocity  is  equal  to  -«.  The  index  of  re- 
fraction of  the  ray  with  vibrations  parallel  to  c  and  advancing  at  right  angles 
to  it,  is  represented  by  7.  Its  velocity  equals  -. 

The  positions  of  the  vibration  axes  vary  in  different  crystals.  In  ortho- 
rhombic  crystals  the  vibration  axes  coincide  with  the  crystallographic  axes. 
In  monoclinic  crystals  one  vibration  axis  coincides  with  crystallographic  b, 
the  other  two  lie  in  the  plane  of  a  and  c  but  are  inclined  to  these  axes  except 
in  one  special  case  where  a  principal  vibration  axis  coincides  with  a  or  c. 
In  triclinic  crystals,  in  general,  no  vibration  axis  coincides  with  a  crystallo- 
graphic axis,  although  in  special  cases  one  axis  of  vibration  may  coincide  with 
one  crystallographic  axis. 

63.  Fletcher's  Indicatrix.1 — As  in  uniaxial  crystals,  so  in  biaxial,  we  may 
represent  the  ease  of  vibration,2  the  refractive  indices,  and  the  ray  and  wave 
surfaces  by  geometrical  figures.  Of  these  the  indicatrix  and  the  ray  and 
wave  surfaces  are  the  most  important. 

Since  the  indices  of  refraction,  in  biaxial  crystals,  differ  in  different  direc- 
tions, and  the  directions  of  maximum  and  minimum  ease  of  vibration  lie  at 
right  angles  to  each  other  (therefore  their  indices  of  refraction  likewise),  and 
a  direction  of  intermediate  ease  lies  at  right  angles  to  the  other  two,  we  may 
represent  the  indices  of  refraction  in  any  direction  in  a  crystal  by  a  triaxial 
ellipsoid.  Thus  we  may  construct  an  ellipsoid  (Fig.  153)  such  that  MA  =  a, 
MB  =  /?,  and  MC  =y.z  In  such  a  figure  there  are  only  three  planes  of  symme- 
try, namely  the  CBC'B',  the  A'CAC',  and  the  A'BABr  planes,  and  these 
are  the  principal  optic  sections  of  the  biaxial  crystal. 

In  a  triaxial  ellipsoid  every  section  is  an  ellipse,  consequently  A'CAC' 
(Figs.  153  and  154)  is  an  ellipse,  and  between  C  and  A  there  will  be  all 
values  of  radii  vectores  between  a  and  7,  the  two  semi-axes.  Since  a  >  b  >  c 
always,  it  is  necessarily  also  true  that  a<  /?  <y,  for  the  values  of  the  velocities 
and  the  indices  of  refraction  are  inversely  proportional.  There  will  be,  for 
the  radius  vector  MP,  some  value  between  MA  (a)  and  MC  (7)  equal  to  ft 
(intermediate  between  a  and  7).  We  will  thus  have,  as  a  section  of  the 
triaxial  ellipsoid,  an  ellipse  whose  semi-axes  are  BM  =  ft  and  MP  =  ft,  that  is, 
a  circle  (Fig.  153).  Likewise,  between  A'  and  C,  there  will  be  a  radius 
vector  equal  to  ft  and,  consequently,  a  radius  of  another  circular  section. 

1  L.  Fletcher:  The  optical  indicatrix  and  the  transmission  of  light  in  crystals,  London, 
1892,  20. 

2  Ease  of  vibration  spheroid,  Fresnel's  surface  d'elasticite   (not  Fresnel's  ellipsoid). 
This  is  a  single  sheet,  triaxial  ellipsoid  whose  equation  is  (x*+yz+z2)  —  (a2#2+  623'2+  cz2)  =o. 

3  It  is  to  be  noted  that  the  directions  of  a,  /?,  and  r  are  not  necessarily  parallel  respec- 
tively to  the  crystallographic  axes  a,  b,  and  c.     <x  may  be  in  the  direction  of  a,  or  b,  or  cy 
and  the  others  likewise. 


ART.  63] 


ANISOTROPIC  MEDIA 


93 


No  other  section  besides  these  two  can  be  circular,  for  any  section  at  a  greater 
or  less  angle  than  that  of  the  MP  section  will  have  a  greater  or  less  value  for 
its  radius  vector  in  the  MCA  plane,  while  in  the  BCB'C  plane  there  can  be 
no  radius  vector  equal  to  the  axis  MA  (  =  «),  which  lies  at  right  angles  to 
the  plane,  for  the  value  of  the  latter  is  less  than  the  minor  axis  of  the  BCB'C' 
ellipse. 


FIG.   153. — A  triaxial  ellipsoid  (indicatrix). 


FIG.   154. — Section  through  the  indicatrix. 


We  can  have,  then,  in  a  triaxial  ellipsoid,  two  sections,  and  two  only, 
which  are  circular.  They  both  contain  the  intermediate  axis  (MB  =  (3)  and 
lie  symmetrically  with  respect  to  the  other  two. 

Every  cross-section  of  the  ellipsoid  represents  the  indices  of  refraction 
of  the  rays  whose  vibrations  are  parallel  to  it  and  whose  wrave  fronts  advance 
normal  to  it.  In  the  circular  sections  the 
difference  between  the  refractive  indices  in 
different  directions  is  zero,  consequently  in 
these  sections  the  velocities  of  the  waves  will 
be  the  same,  and  they  will  suffer  no  double 
refraction.  The  normals  ATA7"  and  N'N'" 
(Figs.  153-154)  are  called  the  primary  optic 
axes  or  optic  binomials.  All  crystals  of 
the  orthorhombic,  monoclinic,  and  triclinic 
systems  have  two  primary  optic  axes,  and 
are,  consequently,  called  biaxial. 

The  index  surfaces  previously  described,  namely  the  sphere  and  the 
ellipsoid  of  rotation,  are  but  special  cases  of  the  triaxial  ellipsoid,  for  if  ft  =  a 
or  ,3=7,  the  figure  will  become  one  of  rotation,  and  if  a  =  ,3=7,  it  will  become 
a  sphere. 


-c 


FIG.   155. — Section  through  the  optical 
indicatrix  of  a  biaxial  crystal. 


94 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  64 


Just  as  in  the  indicatrix  of  uniaxial  crystals,  so  also  in  that  of  biaxial 
crystals  does  Mr  (Fig.  155)  represent  the  direction  of  propagation  of  the  ray, 

and          its  velocity;  consequently  Mr  = 


Equation  of  the  indicatrix.     Analytically  we  have  the  general  equation  of  a 
triaxial  ellipsoid  whose  semi-axes  MA  —  «,  MB  =  P,  and  MC  =  y: 


iii. 
or,   substituting  a  =  — ,  b  =  ^,  c=-,  we  have 


(2) 


64.  The  Ray  Surface  of  a  Biaxial  Crystal. — We  can  develop  the  ray  sur- 
face of  a  biaxial  crystal  in  the  same  way  that  we  developed  the  ray  surface  of 
a  uniaxial  crystal. 


. 


z 

FIG.   156.  FIG.  157. 

FIGS.  156  AND  157. — The  indicatrix  and  a  section  through  the  ray  surface  along  the  MZX  plane. 
Scale  of  indicatrix  one-sixth  of  the  scale  of  the  ray  surface. 

Let  Fig.  156  represent  part  of  the  indicatrix  of  a  biaxial  crystal  in  which 
7>j8>o:.  For  convenience  of  drawing  let  MZ=y  =  4,  MY=P  =  3,  and  MX 
=  a=2.  These  are  the  indices  of  refraction  along  the  three  principal  axes 
and  are  called  the  principal  indices  of  refraction. 

Let  us  consider  first  the  rays  vibrating  in  the  plane  MZX,  Fig.  157.  From 
the  point  M  two  rays  will  travel  in  the  direction  of  X,  one  of  which  has  its 
vibration  direction  parallel  to  MZ.  Now  the  index  of  refraction  of  the  ray 
which  vibrates  parallel  to  MZ  is  7  (Fig.  156),  consequently  the  velocity 


ART.  64]  ANISOTROPIC  MEDIA  95 

of  the  ray  advancing  in  the  direction  of  X  will  be  -  because  v  =  -.     In  a  unit 

of  time,  with  the  values  assumed  above,  since  »*  =  ->  the  distance  traveled 

by  the  ray  will  be  Ma  =  1/4.  (In  Fig.  157,  6  cm.  is  taken  as  unity,  conse- 
quently Ma  =  1/4  =  15  mm.)  The  other  ray  advances  from  M  toward  X 
with  vibrations  parallel  to  M  Y,  with  an  index  of  refraction  equal  to  /3,  and 

therefore  a  velocity  of  „>  and  travels  a  distance  equal  to  «.     When/  =  i,  the 
P  P 

distance  equals  ^  or  20  mm. 

At  the  same  time  that  the  rays  of  light  are  traveling  from  M  to  X,  other 
rays  travel  from  M  toward  Z.  One  ray,  with  vibrations  parallel  to  M X, 

will  have  an  index  of  refraction  of  a  and  will  advance  a  distance  of  Me'  =  — 

in  the  direction  of  Z.  Mc'  =  —  =  1/2  =30  mm.  The  second  ray,  in  the  same 
direction,  will  have  its  vibrations  parallel  to  MY.  Its  index  of  refraction 
being  /3,  the  distance  traveled  in  a  unit  of  time  will  be  •«=  1/3  =  20  mm.  (Me). 

In  the  directions  MX'  and  MZ'  the  two  rays  will  advance  the  same 
amounts  but  in  opposite  directions  from  MX  and  MZ. 

In  any  intermediate  direction  in  the  plane  MXZ,  such  as  Mrr',  one  ray 
will  have  its  vibrations  constantly  parallel  to  MY,  irrespective  of  the  direc- 
tion of  its  transmission.  Its  velocity  will  be  uniformly  «  and  the  distance 

traveled  in  a  unit  of  time  will  be  «  =  1/3  =  20  mm.     Since  the  velocity  is  the 

same  in  every  direction  in  the  plane  MXZ,  the  ray  front  will  be  a  circle. 
The  other  ray,  whose  vibrations  lie  in  the  plane  MXZ,  will  advance  with  a 
velocity  varying  as  the  radii  vectores  of  the  ellipse  whose  major  and  minor 
semi-axes  are  7  and  a.  The  vibrations,  being  always  parallel  to  the  tangent 
to  the  ellipse  at  the  extremity  of  the  ray  (cf.  Art.  54)  will  be  at  right  angles  to 
the  ray  only  along  the  axes  MX  and  MZ. 

Completing  the  ray  front  for  the  plane  XZX'Z'  we  will  have  two  curves; 
a  circle,  representing  the  front  of  the  rays  whose  vibrations  are  at  right 
angles  to  the  plane  of  reference,  and  an  ellipse,  representing  the  front  of  the 
rays  whose  vibrations  lie  within  that  plane.  The  vibrations  in  the  latter  are, 
in  general,  not  at  right  angles  to  the  ray. 

In  like  manner  in  the  plane  MYZ  (Fig.  156),  which  is  represented  as 
rotated  into  the  plane  of  the  paper  in  Fig.  158,  there  will  be  two  rays  from  M 
advancing  toward  Y.  One,  with  vibrations  parallel  to  MZ  and  an  index 

of  refraction  of  f,  will  advance  with  a  velocity  of  -,  equal,  in  the  case  cited, 
in  a  unit  of  time,  to  a  distance  of  Mb  =1/4  =  15  mm.  The  other,  with  vibra- 


96 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  64 


tions  parallel  to  MX  and  an  index  equal  to  a,  will  advance  with  a  velocity 
equal  to  —  =  1/2,  or  a  distance,  in  a  unit  of  time,  of  Mb' =  1/2  =  30  mm. 


i*       x 


z'  Y 

FIG.    158..  FIG.   159. 

FIGS.   158  AND  159. — Sections  through  the  ray  surface  along  the  MYZ  and  MXY  planes.     Scale  same 

as  Fig.  157. 

In  the  direction  of  MZ  one  ray  will  have  its  vibrations  parallel  to  M Y,  its 
index  of  refraction  equal  to  j8,  its  velocity  «,  and  its  distance  of  transmission 


FIG.   160.  FIG.   161. 

FIGS.   160  AND  161. — Form  of  the  ray  surface  developed  on  a  hinged  blackboard. 


-Q,  which  equals  1/3,  in  a  unit  of  time,  or  20  mm.  (Me)  on  the  scale  adopted  in 
the  figures.     The  other  ray  vibrates  parallel  to  MX,  has  an  index  of  a,  a 


ART.  64]  AXISOTROPIC  MEDIA  97 

velocity  of  -,  and  travels  a  distance  of  1/2  or  30  mm.  (Me').  In  any  interme- 
diate direction,  as  Mss',  there  will  likewise  be  two  rays.  The  one  with  vibra- 
tions perpendicular  to  the  plane  M YZ  will  advance  with  an  index  of  a  and  a 

velocity  of  — ,  and  this  velocity  will  be  uniform  in  every  direction  in  the  sec- 
tion, since  its  vibrations  remain  parallel  to  the  same  line,  consequently  its 
ray  front  will  be  a  circle.  The  other  ray,  with  vibrations  lying  in  the  plane 
MZY,  will  have  velocities  varying  as  the  radii  vectores  of  the  ellipse  having 
axes  of  7  and  0. 

In  the  horizontal  plane,  MYX  of  Fig.  156,  which  is  represented  as  rotated 
into  the  plane  of  the  paper  in  Fig.  159,  all  of  the  rays  having  vibrations  parallel 

to  MZ  will  advance,  in  a  similar  manner,  a  distance  equal  to  -  =  1/4  =  15  mm. 

(Ma  and  Mb).  The  ray  front,  consequently,  is  a  circle.  The  rays  whose 
vibrations  lie  within  the  XY  plane  will  advance  with  different  velocities,  and, 
consequently,  different  distances  in  different  directions  in  the  plane.  The 
ray  advancing  along  MX  will  have  vibrations  parallel  to  M  Y,  a  refractive 

index  equal  to  £,  a  velocity  of  «,  and,  in  the  time  -«,  will  travel  a  distance  of 

o,  equal,  in  a  unit  of  time,  to  g  =1/3  =  20  mm.  (Maf).     Intermediate  rays, 

as  before,  will  reach  the  ellipse  whose  semi-diameters  are  a  and  0. 

The  form  of  the  ray  surface  may  be  made 
clearer  by  working  it  out  on  three  planes  at  right 
angles  to  each  other,  as  shown  in  perspective  in 
Figs.  160  and  161.  A  blackboard,  hinged  at  the 
joints,  or  even  part  of  a  cigar  box,  may  be  used. 
Vibration  directions,  perpendicular  to  the  plane, 
are  shown  by  pins,  while  vibration  directions 
lying  in  the  plane  are  shown  by  marks  on  the 
board. 

If,  now,  we  consider  the  form  of  the  solid 
which  has  been  developed,  we  will  see  that  it 
differs  decidedly  from  the  symmetrical  ellip-  ftc-  '^.-Piaster  model  of  a  bi- 

.     .  .  .  L  axial  ray  surface. 

soid  of  rotation  of  uniaxial  crystals.     It  is  a 

warped  surface  such  as  is  shown  in  Fig.  1-62,  symmetrical  along  the  three 

principal  axes  and  having  four  depressions  lying  in  a  single  plane. 

The  equation  of  the  ray  surface  of  a  biaxial  crystal.1 

The  form  of  the  ray  surface  may  be  expressed  by  the  equation 

yf         *  ,0 

r'1-  cr       r2-b2       r2-c- 

1  L.  Fletcher:  The  optical  indicatrix,  etc.,  p.  37,  §7. 
7 


98 


MANUAL  OF  PETROGRAPHIC  METHODS 


or,  substituting  r2  =  #2+3/2-|-z2,  we  have 


'•z2  -^  + 


=  1. 


[ART.  6S 


(4) 


Equation  of  the  velocity  of  any  ray  in  a  biaxial  crystal.1 

The  equation  of  the  velocity  of  any  ray  (r,  Fig.  155)  whose  normal  NR  inter- 
sects the  indicatrix  at  a  point  represented  by  x\,  y\,  and  z\  is 


This  is  the  equation  of  any  radius  vector  of  the  ray  surface  corresponding  in  direction 
with  the  line  Mr  of  the  indicatrix. 

65.  The  Wave  Surface  of  a  Biaxial  Crystal.  —  We  saw,  in  the  develop- 
ment of  the  wave  surface  of  a  uniaxial  crystal,  that  it  differed  slightly  from 
the  ray  surface.  In  the  former  the  surface  is  not  an  ellipsoid  of  rotation,  and 
in  the  latter  it  is.  By  the  same  construction  we  may  develop  the  wave  surface 
of  a  biaxial  crystal. 


z' 

FIG.   163.  FIG.   164.  FIG.   165. 

FIGS.   163  TO  165. — Principal  sections  through  the  wave  surface. 

Let  the  dotted  lines,  Fig.  163,  represent  a  principal  section  through  the 
ray  surface  along  the  MXZ  plane.  The  ray  Mr',  with  vibrations  within  the 
plane,  will  reach  the  ray  surface  at  /.  The  wave  front  of  all  rays  progressing 
parallel  to  Mr'  will  lie  along  the  tangent  r'n'  (cf.  Art.  52),  consequently  the 
intersection  of  the  normal  Mn'  with  the  tangent  at  n'  will  be  a  point  on  the 
wave  surface.  The  whole  curve  of  wave  fronts,  shown  as  a  solid  line  in 
Fig.  163,  may  thus  be  constructed.  For  any  ray  Mr,  whose  vibrations  take 
place  at  right  angles  to  the  MXZ  plane,  the  direction  of  the  ray  and  the  normal 
to  the  tangent  coincide  (cf.  Art.  55),  and  the  wave  front  for  those  rays  is  a 
circle. 

Likewise,  in  the  planes  MYZ  and  MXY,  the  curve  of  wave  fronts  may 
be  constructed  as  shown  in  Figs.  164  and  165.  The  solid  resulting  from  all  of 

1  Idem:  Ibidem,  p,  36,  §4. 


ART.  66] 


ANISOTROPIC  MEDIA 


99 


the  wave  fronts  is  similar  in  form  to  the  ray  surface,  but  does  not  coincide 

with  it. l 

The  equation  of  the  wave  surface  of  a  biaxial  crystal.2    Analytically  the  ray  sur- 
face may  be  expressed  by  the  equation 


or,  substituting  *2+;y2+22=r2,  we  have, 


2+z2-  a2 


-  b2 


-  2 


(7) 


66.  Optic  Biradials  or  Secondary  Optic  Axes. — Let  us  consider  a  little  more 
fully  both  ray  and  wave  surfaces.     If  we  examine  Fig.  157  we  shall  see  that 


FIG.  166. — Section    through    one- 
fourth  of  the  ray  surface. 


M 


FIG.  167. — Section  through  one- 
fourth  of  the  wave  surface. 


there  are  four  points,  p,  p',  p",  and  p"f,  where  the  two  ray  fronts  inter- 
sect. Since  these  curves  represent  the  velocities  of  the  rays,  obviously  along 
the  lines  p'Mp  and  p"Mp"r  (Fig.  157),  within  the  crystal,  the  rays  will  travel 
together  without  double  refraction.  Upon  emerging,  however,  the  waves 
advance  normal  to  the  tangents  to  the  wave  surfaces.  Now  the  tangent  to 
the  circle  at  p  (Fig.  166)  is  //',  while  the  tangent  to  the  ellipse  is  e'e",  and  the 
two  waves,  not  having  a  common  front,  are  doubly  refracted  upon  emerging, 
consequently  two  light  waves  advance  in  the  directions  po  and  pe.  The  lines 
Mp,  Mp',  etc.,  along  which  the  two  rays  advance  with  equal  velocity,  are  called 
the  secondary  optic  axes,  optic  biradials,3  or  lines  of  single  ray  velocity.4 

1  An  illustration  of  a  plaster  model  of  the  wave  surface  is  given  in  Rosenbusch-Wiilnng, 
Mikroskopische  Physiographic,  I-i,  93. 
2L.  Fletcher:  Op.  cit.,  60,  §39. 

3  Idem:  Op.  cit.,  43-44. 

4  Sir  William  Hamilton:  Third  supplement  to  an  essay  on  the  theory  of  systems  of  rays. 
Read  Jan.  23  and  Oct.  22,  1832.     Trans.  Roy.  Irish  Acad.,  Dublin,  XVII  (1837),  1-144, 
in  particular,  132. 


100  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  67 

67.  Optic  Binomials  or  Primary  Optic  Axes. — We  have  already  seen  that 
in  the  optical  indicatrix  there  occur  two  sections  which  are  circular  (Art.  63), 
and  for  each  ray  having  vibrations  parallel  to  it  a  wave  will  advance  in  a  di- 
rection at  right  angles  to  it.  Identical  in  direction  with  the  normals  to  the 
circular  sections  are  certain  lines  in  the  wave  surface.  If  we  examine  the 
MXZ  section  of  the  wave  surface  (Fig.  167),  we  will  see  that  the  only  place 
where  two  waves,  with  vibrations  at  right  angles  to  each  other,  coincide,  is 
where  the  oval  crosses  the  circle  at  n'.  This  point  is  located  by  the  normal 
to  the  wave  front  which  is  produced  by  the  ray  advancing  in  the  direction 
of  r,  and  therefore  represents  the  direction  of  transmission  of  the  wave  pro- 
duced by  that  ray.  The  particular  ray  with  which  we  are  concerned  is  the 
one  whose  front  is  tangent  to  the  circle  as  well  as  to  the  ellipse.  In  other  words 
it  is  the  ray  which  causes  a  wave  to  advance  along  the  same  line,  and  with  the 
same  velocity,  as  the  wave  produced  by  another  ray  whose  vibrations  take 
place  at  right  angles  to  the  section. 

But  a  plane,  tangent  to  the  ray  fronts  at  r'  and  n'  (Fig.  167),  will  also  be 
tangent  to  the  ray  surface  (Fig.  162)  in  other  points,  namely,  in  a  continuous 
circle1  having  a  diameter  of  r'n' .  Since  the  wave  normal  Mn'  forms  a 
right  angle  with  the  tangent  r'n',  the  latter  being  the  trace  of  the  base  of 
the  cone  r'Mn'  whose  apex  is  at  M,  it  follows  that  all  rays  refracted  from  M 
to  r'  or  to  any  other  point  lying  in  the  periphery  of  the  base  of  the  cone,  must  be 
represented  by  a  wave  advancing  along  the  common  normal  Mn' .  The  line  Mn' , 
consequently,  represents  the  only  line  along  which  more  that  one  wave  ad- 
vances, and  is,  therefore,  at  right  angles  to  the  circular  section  of  the  indicatrix. 
The  normal  to  the  circular  section  and  the  normal  to  the  wave  fronts  coincide. 
It  is  the  primary  optic  axis,  also  called  the  optic  binormal,2  line  of  single 
normal  velocity,3  or  axis  of  single  wave  velocity. 

In  all  the  preceding  figures  showing  indices  of  refraction  or  velocities  of 
light,  the  differences  in  different  directions  have  been  greatly  exaggerated 
over  those  which  occur  in  nature.  As  a  matter  of  fact,  the  ellipses  actually 
do  not  depart  greatly  from  circles,  consequently  the  difference  between  the 
ray  surface  and  the  wave  surface  of  a  crystal  is  not  great.  Likewise  the  pri- 
mary and  secondary  optic  axes,  represented  by  Mn  and  Mp,  Fig.  167,  nearly 
coincide,  the  difference  between  their  directions  being  rarely  over  one  degree. 
When  simply  optic  axes  are  mentioned,  the  primary  optic  axes  are  usually 
understood. 

68.  Interior  Conical  Refraction. — From  the  sections  of  the  ray  surface 
cut  by  the  three  principal  planes,  Figs.  157,  158,  and  159,  we  saw  that  each 
consists  of  an  ellipse  and  a  circle  having  the  same  center,  but  that  in  only  one, 
the  XZ  plane  (Fig.  157)  do  the  two  intersect  in  four  points,  p,  p',  p",  p'". 

1  See  Article  68. 

2L.  Fletcher:  Op.  cit.,  62-63. 

3  Sir  William  Hamilton:  Op.  cit.,  132. 


ART.  OS]  AMSOTROPIC  MT.DJA  101 

Above  each  of  these  points  a  tangent  to  both  the  circle  and  the  ellipse  may 
be  drawn,  as  shown  in  one  quadrant  in  Fig.  166,  r'n1.  Now  these  tangents 
are  the  traces  of  planes  which  not  only  touch  the  wave  surface  at  the  points 
r'  and  »',  but  in  a  continuous  line,  which  was  shown  by  Sir  William  Hamil- 
ton1 to  be  a  circle. 

We  will  thus  have  formed  an  oblique  cone  having  a  circular  base  whose 
diameter  is  r'n',  and  an  altitude  of  n'M .     Not  only  is  n'M  the  altitude,  but 
it  also  forms  one  of  the  lines  of  the  cone  extending  from  the  apex  to  the  cir- 
cumference of  the  base.     Since  n'M  not  only  represents  the  direction  of  trans- 
mission of  the  wave  produced  by  the  ray  Mr',  but  also  of  those  produced  by 
all  other  rays  progressing  from  M  to  any  point  on 
the  ray  surface  where  this  is  touched  by  the  tan- 
gent plane  (contact  a  circle),  the  sum  of  all  these 
rays   will  represent  the  curved  surface  of  a  cone 
whose  base  is  a  circle  with  a  diameter  of  r'n'  and 
whose  altitude  is  n'M.     If,  then,  a  section  be  cut 
from  a  biaxial  crystal  so  that  the  two  faces  are  at 
right  angles  to  the  line  n'M  and  a  beam  of  light 
be  made  to  enter  at  M,  it  will  pass  through  the    refraction. 
crystal  as  the  cone  r'Mn'  and  emerge  as  a  cylinder 

with  circular  cross  section  r'r",  n'n" .  If,  on  the  other  hand,  the  beam  has 
a  diameter  of  r'n' ',  and  enters  from  without,  it  will  converge  to  the  apex  at 
M.  This  property  of  biaxial  crystals  is  called  interior  (or  internal)  conical 
refraction. 

This  cone  of  light  was  shown  experimentally  by  Lloyd2,  who  passed  a 
fine  beam  of  light  along  the  optic  axis  (Mn,  Fig.  168)  of  a  plate  of  aragonite. 
Two  thin  metal  screens,  /  and  //,  pierced  by  small  holes,  one  screen  at  a 
little  distance  and  the  other  in  contact  with  the  plate,  were  used  to  regulate 
a  narrow  beam  of  light,  and  the  emerging  ray  was  examined  on  the  screen 
///.  When  the  angle  of  incidence  was  other  than  that  required  to  refract 
the  one  ray  along  the  optic  axis  Mn,  two  spots  of  light  were  seen  upon  the 
screen.  The  crystal  was  moved  very  slowly,  and  the  instant  the  light  fell 

1  Sir  William  Hamilton:  Op.  dL 

See  also  Th.  Liebisch:  Physikalische  Krystallographie,  Leipzig,  1891,  341-345. 

2  Rev.  Humphrey  Lloyd:  On  the  phenomena   presented  by  light  in  its  passage  along  the 
axes  of  biaxial  crystals.     Phil.  Mag.,  II  (1833),  112-120. 

Idem:  Ueber  die  Erscheinungen  beim  Durchgange  des  Lichts  durch  zweiaxige  Krystalle 
Idngs  dcren  Axen.  Translation  of  preceding.  Pogg.  Ann.,  XXVIII  (1833),  91-104. 

Idem:  Further  experiments  on  the  phenomena  presented  by  light  in  its  passage  along  the 
axes  of  biaxial  crystals.  Phil.  Mag.,  II  (1833),  207-209. 

Idem:  Fernere  Versuche  iiber  die  Erscheinungen  beim  Durchgange  des  Lichts  durch 
zweiaxige  Krystalle  Icings  deren  Axen.  Translation  of  preceding.  Pogg.  Ann.,  XXVIII 
(1833),  104-108. 

Idem:  On  the  phenomena  presented  by  light  in  its  passage  along  the  axes  of  biaxial  crystals. 
Read  Jan.  28,  1833.  Trans.  Roy.  Irish  Acad.,  Dublin,  XVII  (1837),  145-157. 


102 


MANUAL  OF^  TETROGRAPHIC  METHODS 


[ART.  69 


at  the  right  angle  of  incidence,  the  two  spots  immediately  united  and  formed 
a  continuous  ring  of  light.  Upon  varying  the  distance  of  screen  ///  from 
the  crystal,  no  enlargement  of  the  ring  was  observed,  showing  the  cylindrical 
form  of  the  emerging  beam.  f 

69.  Exterior  Conical  Refraction. — If  a  section  be  cut  from  a  biaxial  crystal 
so  that  the  two  parallel  faces  are  normal  to  the  line  Mp  (Fig.  166),  and  a  ray 
of  light  be  passed  along  the  line  Mp,  it  will  emerge  in  the  cone  formed  by  the 
perpendiculars  to  the  planes  tt'  and  e'e".     Conversely;  a  cone  of  light  ope, 
entering  along  the  secondary  optic  axis,  will  pass  through  along  the  single 
line  pM .     This  phenomenon  is  called  exterior  (or  external)  conical  refraction 
and  also  was  shown  experimentally  by  Lloyd. 

70.  Optic  Axial  Angle,  True  and  Apparent. — As  we  have  already  seen 
(Fig.  154),  if  a  circle,  having  a  radius  of  /3,  is  drawn  with  M  as  a  center,  the 


FIG.   169. — Optic  axial  angle  and  bisectrices.        FIG.   170. — True  and  apparent  optic  axial  angle. 

lines  connecting  M  with  the  point  where  the  circle  cuts  the  ellipse,  represent 
the  traces  of  the  two  circular  sections.  Perpendiculars,  MN  and  MNf, 
erected  to  this  plane,  represent  the  optic  axes  of  the  crystal. 

Let  Fig.  169  represent  a  section  through  the  indicatrix  of  a  biaxial  crystal, 
and  let  M Z  =y,  MX  =  a,  and  Mp=$.  Then  Mp  and  Mp'  will  represent  the 
traces  of  the  circular  sections,  and  Mn  and  Mnf,  normal  to  these  planes,  the 
primary  optic  axes.  One  optic  axis  coincides,  in  direction,  with  the  line 
Mn'  of  Figs.  1 66  and  167. 

The  angle  n'Mn  (Fig.  169)  between  the  two  optic  axes  is  called  the  optic 
angle  or  axial  angle.  It  is  indicated  by  the  symbol  2V,  and  is  the  true 
optic  angle  in  contradistinction  to  the  apparent  optic  angle  in  air,  which  is 
indicated  by  the  symbol  2E.  The  relation  between  the  two  is  shown  in  Fig. 
170  in  which  AOC  is  2  V,  and  A'O'C',  the  apparent  angle  in  air,  is  2E.  Some- 
times the  optic  angle  is  determined  by  immersion  in  water,  oil,  etc.  In  this 
case  the  angle  is  indicated  by  2H. 


ART.  71]  ANISOTROPIC  MEDIA  103 

71.  Equations  Expressing  the  Value  of  the  True  Axial  Angle.  —  Analytically 
the  optic  angle  may  be  computed  if  the  values  of  the  refractive  indices  are  known. 
The  equation  of  the  indicatrix  (Eq.  i,  Art.  63)  is 

-&+Ji+-*  =  i-  (J) 

For  any  point  in  the  elliptical  section,  as  p,  Fig.  169,  y  =  o,  and 


The  equation  of  the  circle  whose  radius  M  p  =  0  is 

*2-hz2=/32,  or  xz=p-z*.  (9) 

Combining  (8)  and  (9)  we  have, 

£2-z2  ,  z2!  ,    T2032-«2) 

--+--  =  i,or22  =  -ii-.  do) 


The  coordinates  of  p  being  x  -and  z,  we  have 

sin  pMX  =  —Q>  or  z—  ft  sin  pMX.  (n) 

Substitute  in  (10), 


or  sin2  />MX  =  -jj^_a^  •  (12) 

But  ^MX  =  9o°-wMAr  =  9o°-F/,  and 

^(Q2_a2\ 

sin2  ^MX  =  sin2  (90°-  TV)  =  cos2F/=™I2.  (13) 


It  is  to  be  noted  that  the  value  here  given  for  Vf  is  one-half  the  acute  optic 
angle  for  a  negative  crystal;  in  other  words,  it  is  the  angle  between  one  optic  axis 
and  the  fast  vibration  direction  (a).  Had  the  other  vibration  axis  (y)  been  chosen, 
the  ormula  would  have  been 


The  equation  of  the  sine  may  be  similarly  expressed.     Equation  (9)  may  be 

written 

Z2=02-*2,  (14) 

which,  substituted  in  (8),  gives 

#2      P2  —  x-  a2(72  —  £2) 

— 2~H 5—""  i,  or**" — „- — 2 —  (15) 

7  y  —  or 


104 


MANUAL  OF  PETROGRAPHIC  METHODS 


But  cos  pM X  =  — '  from  which  x=  0  cos  (90°—  F/)  =  ft  sin  F/. 
& 


[ART.  72 
(16) 


Substitute  in  (15) 


^sin2^    3-J 


or 


•        g      TT       _      "      Vr      —V)  ,  v 

/—  /?2(72— a2)'  ^*" 

As  above,  the  equation  for  the  angle  between  the  optic  axis  and  the  slow  vibra- 
tion direction  is 


Sin2   y   =T-«      . 

/32(72-«2) 


The  tangent  relation  may  be  obtained  from  the  equations 

sin  Vr  sin2F/ 

tan  F/= ~.  and  tan2F/  = 

cosF/  cos2F/ 

Substituting  equations  (13)  and  (14)  in  (18),  we  have 


(i7a) 


(18) 


(19) 


Likewise  tan2  Vs=    z,  2        ^.  (191 

72.  Relation  between  the  True  and  Apparent  Axial  Angles.— In  Fig.  171  let 
F  =  n'Mb  =  Mric  —  i  (the /angle  of  incidence), 
E  =  n'm'b  =  m'nfd  =  en'f=r  (the  angle  of  refraction). 


flsv 


A.  Xr 


FIG.   171. — Relation  between  true  and  apparent  axial  angle. 

In  passing  from  a  denser  to  a  rarer  medium  we  have  (Art.  41). 


sm  i  __  i 
sin  r~  n 


Substitute  /S,  the  mean  refractive  index,  for  n,  and  we  have 


sin  F      i 

~  ---=;  =  -,  or  sin 

sin  E      ft- 


sin  F. 


(20) 


ART.  75]  ANISOTROP1C  MEDIA  105 

That  is,  the  sine  of  the  true  optic  angle  of  the  mineral  multiplied  by  the 
intermediate  index  of  refraction  will  give  the  sine  of  the  apparent  axial  angle  in 
air. 

73.  Plane  of  the  Optic  Axes. — It  is  obvious,  from  the  statements  made  in 
Articles  63  and  67,  that  the  optic  axes  must  always  lie  in  the  plane  of  maximum 
and  minimum  indices  of  refraction  (7  and  a),  resp.  velocities  (a   and  c), 
consequently  the  rule  follows  that  the  plane  of  the  optic  axes  is  the  plane 
containing  y  and  a,  resp.  C  and  a. 

74.  Bisectrices. — The  lines  which  bisect   the  angles  between  the  optic 
axes  are  known  as  the  bisectrices.     That  bisecting  the  acute  angle  is  called 
the  acute  bisectrix,  while  that  bisecting  the  obtuse  angle  is  called  the  obtuse 
bisectrix.     They  are  expressed  by  the  symbols  Bxa  and  Bx0.    The  bisec- 
trices always  lie  at  right  angles  to  each  other,  and  always  coincide  with  the 
axes  of  minimum  and  maximum  ease  of  vibration. 

75.  Positive  and  Negative  Biaxial  Crystals. — We  found,  in  uniaxial  crys- 
tals, that  as  the  c  axis  coincided  with  the  slowest  or  fastest  ray,  the  crystals 
were  called  positive  or  negative.     Biaxial  crystals  are  considered  positive 
when  the  acute  bisectrix  coincides  with  the  direction  of  vibration  of  the  slow 
ray  (c),  and  negative  when  it  coincides  with  the  vibration  direction  of 
the  fast  ray  (a). 

There  are  certain  crystals  in  which  the  acute  bisectrix,  for  example,  is 
the  vibration  direction  of  the  fast  ray,  and  the  crystal  is,  consequently,  nega- 
tive. By  a  progressive  change  in  chemical  composition  there  may  be  formed 
other  minerals  of  the  same  group.  Coincident  with  this  change  in  composi- 
tion, the  acute  optic  angle  may  become  larger  and  larger  until  it  reaches  90°, 
beyond  which  point  the  acute  bisectrix  lies  in  the  direction  of  the  vibrations 
of  the  slow  ray,  and  the  mineral  is  positive.  Such  a  change,  for  example, 
takes  place  in  the  hypersthene-enstatite  group.  Hypersthene,  (Mg,Fe)SiO3, 
with  an  axial  angle  of  about  80°  and  the  fast  vibrations  in  the  direction 
of  the  acute  bisectrix,  is  negative,  while  enstatite,  MgSiOa,  with  an  axial 
angle  of  70°  and  the  slow  vibrations  in  the  direction  of  the  acute  bisectrix,  is 
positive.  That  is,  with  the  decrease  in  the  percentage  of  iron,  the  angle  has 
changed  from  80°  to  110°.  Intermediate  between  these  two  there  are  other 
orthorhombic  pyroxenes  with  varying  proportions  of  iron,  consequently 
having  axial  angles  which  lie  between  80°  negative  and  70°  positive.  At 
some  point  between  the  two  the  axial  angle  is  90°,  but  90°  only  for  a  certain 
color  of  light.  The  effect  of  different  colored  light  is  well  shown  in  danburite, 
between  the  optic  axes  of  which  there  is  an  angle  of  89°  14'  by  green 
light,  and  90°  24'  by  blue.  That  is,  the  mineral  is  negative  for  green  and 
positive  for  blue. 


106 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  76 


When 


=  9O°, 


=  45°,  and  tan2F  =  i,  equation  (19)  becomes 


and 


For  any  other  value  of  0  the  mineral  will  be  either  positive  or  negative. 
Since  the  wave  surface  is  actually  nearly  a  sphere,  one  may  say,  with  approxi- 
mate truth,  that  the  nearer  /?  approaches  a,  the  nearer  the  optic  axes  will  lie 
to  the  vibration  direction  oj  C,  and  vice  versa,  consequently  for  all  values  except 
close  to  7  =  45°,  if  7— /3>/3—  a  the  mineral  is  positive,  and  if  7  —  /3</3—  a  the 
mineral  is  negative. 


FIG.  172.  FIG.  173.  FIG.  174- 

FIGS.  172  TO  174. — Tourmaline  crystals  in  parallel  position  permitting  light  to  pass  through  (172), 
crossed  and  with  light  extinguished  (173),  and  at  some  angle  between  o°  and  90°,  permitting  some 
of  the  light  to  pass  through  (174)* 

76.  Polarization  by  Double  Refraction. — The  statement  was  made  that 
when  a  ray  of  light  passes  into  an  anisotropic  medium,  it  is  resolved  into  two 
rays;  that  is,  it  is  polarized  by  double  refraction  into  two  sets  of  waves 

vibrating  in  planes  at  right  angles  to  each 
other.  This  can  be  shown  very  readily  in  the 
case  of  certain  doubly  refracting  crystals  which 
naturally  absorb  one  set  of  vibrations.  For 
example,  if  a  transparent  crystal  of  tourmaline 
is  cut  parallel  to  crystallographic  c,  and  is 
placed  over  another  crystal  of  the  same  mineral 
similarly  cut,  it  will  be  found  that  when  the  two 
c  axes  are  parallel,  the  light  will  pass  through 
FIG.  z 75.— Resolution  of  polar-  (-pig.  172),  but  when  they  lie  at  right  angles,  it 

ized    light    upon     passing    through      V      f  .          J  .  to        ' 

two  crystals  of  tourmaline.  will  be  totally  extinguished.     It  is  as  though  the 

crystals  were  made  up  of  gratings  of  parallel  bars 

and  the  polarized  light  was  a  sheet  of  cardboard.  When  the  two  crystals 
are  in  parallel  position  the  sheet  slips  through  without  hindrance,  but 
when  they  are  crossed  the  passage  is  closed  (Fig.  173).  If  instead  of 
placing  the  two  crystals  at  right  angles,  they  are  turned  to  some  other 
angle,  an  amount  of  light  proportional  to  the  rotation  passes  through. 


ART.  77]  ANISOTROPIC  MEDIA  107 

In  Fig.  175  let  OP  represent  the  amplitude  of  the  vibrations  of  the  light 
passing  through  the  horizontal  crystal.  When  it  enters  the  inclined  crystal 
it  is  resolved  into  two  rays,  one  of  which  vibrates  parallel,  and  the  other  at 
right  angles  to  crystallographic  c  (OA  and  OB).  Of  these  two  rays  only  the 
one  parallel  to  crystallographic  c  (OA)  can  pass  through  tourmaline,  the  other 
(OB)  being  absorbed,  and  the  amplitude  of  the  vibrations  which  reach  the 
eye  is  less  than  that  with  which  it  arrived  at  the  under  side  of  the  upper 


FIG.   176. — Tourmaline  tongs  (Central  Scientific  Co.,  Chicago). 

crystal.  As  the  upper  crystal  is  rotated  more  and  more  toward  a  position  at 
90°  to  the  horizontal  crystal,  the  OA  component  becomes  smaller  and 
smaller,  therefore  less  and  less  light  reaches  the  eye  until,  when  the  90° 
position  is  reached,  there  is  complete  darkness. 

The  tourmaline  tongs  (Fig.  176),  which  are  used  to  determine  whether  a 
mineral  is  singly  or  doubly  refracting,  are  based  on  this  principle.  They 
were  formerly  much  used  in  the  determination  of  gems. 

77-  Circular  and  Elliptical  Polarization.— We  have  spoken,  so  far,  only 
of  light  polarized  in  planes,  consequently  called  plane  polarized,  but  it  may 
be  polarized,  also,  in  circles  and  ellipses. 

Let  A  and  A'  (Fig.  177)  represent  two  rays  of  light 
which  meet  the  doubly  refracting  crystal  MNOP  at  B 
and  Bf.  Each  ray  is  broken  into  two  components  (BC, 
BD,  and  B'C,  B'E)  with  vibrations  at  right  angles  to 
each  other.  Let  the  two  rays  A  and  A '  be  so  selected 
that  the  extraordinary  component  of  one,  and  the  ordi- 
nary  component  of  the  other,  emerge  at  the  same  point 
C.  On  entering  the  crystal  at  B  and  B',  the  two  rays  were  in  the  same  phase, 
but  since  the  ray  B'C  has  traveled  a  longer  path  than  the  ray  BC,  they  are 
no  longer  so  on  emergence.  We  thus  have  two  rays  emerging  at  C,  with  vibra- 
tions in  planes  at  right  angles  to  each  other  and  in  slightly  different  phase.  As 
we  have  already  seen  (Art.  27),  an  elliptical  motion  is  set  up  in  such  a  case. 
Added  to  this,  however,  is  the  forward  movement  from  C  to  X,  and  as  a  result 
the  complete  motion  is  in  the  form  of  a  helix  of  elliptical  cross-section.  Light 
of  this  kind  is  said  to  have  elliptical  polarization. 

If  the  two  amplitudes  are  equal  and  the  phasal  difference  is    -  -  or  - 

4         4 
(Art.  27),  the  resulting  motion  is  circular,  and  we  have  circular  polarization.1 

1  Care  should  be  taken  not  to  confuse  circular  polarization  \vith  rotary  polarization 
(Art.  78),  which  is  something  quite  different. 


108  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  78 

78.  Rotary  Polarization. — In  1811  Arago1  discovered  that  if  a  thick 
plate  of  quartz,  with  parallel  faces  cut  at  right  angles  to  the  optic  axis,  be 
examined  in  plane  polarized  light  between  crossed  nicols,  it  does  not  appear 
dark,  as  one  would  expect  of  a  uniaxial  crystal,  but  shows  an  interference  tint. 
If  the  section  be  rotated  on  the  stage  of  the  microscope,  this  color  does  not 
change,  and,  if  monochromatic  light  be  used,  it  is  only  by  rotating  one  or 
the  other  nicol  through  a  certain  angle  that  darkness  can  be  produced.  It 
is  perfectly  clear,  therefore,  that  the  emerging  light  is  plane  polarized  but 
not  in  the  direction  in  which  it  entered  the  crystal.  In  other  words,  the  plane 
of  polarization  has  been  rotated. 

The  only  other  natural  mineral  known  to  possess  the  property  of  rotary 
polarization  is  cinnabar.  The  effect  is  rather  weak  in  each  case.  In  the 
former  it  is  about  1/200  as  great  as  the  double  refraction  in  a  direction  at 
right  angles  to  the  axis,  in  the  latter  about  thirteen  times  as  great  as  in  quartz. 
Rotary  polarization  is  found,  however,  in  many  organic  substances  which 
crystallize  in  enantiomorphous  forms. 

All  substances  which  possess  the  power  of  rotating  the  plane  of  polariza- 
tion are  called  active,  the  others  inactive. 

While  rotary  polarization  is  of  interest,  it  is  of  no  great  use  in  petrography 
except  that  advantage  is  taken  of  it  in  the  construction  of  certain  sensitive 
plates,  such  as  that  of  Biot,  Klein,  etc. 

The  amount  of  rotation  depends  upon  the  color  of  the  light  used  and  the 
thickness  of  the  section,  as  was  shown  by  Biot.2  In  the  table  on  the  following 
page  the  values  in  millimeters  for  the  rotation,  given  in  the  fourth  column, 
were  determined  experimentally  by  Soret  and  Sarasin.3  They  may  be  cal- 
culated by  Lommel's4  formula, 

1  F.  J.   Arago:  Memoir e  sur  une  modification  remarquable  qu'eprovent  les  rayons  lumi- 
neux  dans  leur  passage  a  traiers  certains  corps  diaphanes  et  sur  quelques  autres  nouieaux 
phenomenes  d'opiique.     Mem.  Acad.  France,  Annee  1811,  Pt.  i,  XII  (1812),  93-134. 

Idem:  Ueber  eine  eigenthumliche  Modification,  welche  die  Lichtstrahlen  beim  Durchgehen 
durch  gewisse  durchsichtige  Korper  erleiden,  und  uber  einige  andere  neue  optische  Erschein- 
ungen.  (Translation  of  the  preceding  by  Gilbert.)  Gilbert's  Ann.,  XL  (1812),  145-161. 

Idem:  Memoire  sur  la  polarisation  coloree.  Oeuvres  completes,  Paris,  X  (n.d.),  36-74, 
especially  54-55* 

2  J.  B.  Biot:  Memoire  sur  un  nouveaux  genre  d' oscillation  que  les  molecules  de  la  lumiere 
eprouiient  en  traiersant  certains  cristaux.     Lu  a  1'Institute,  31  mai,  1813,  et  3  nov.,  1813. 
Mem.  Acad.  France,  Annee  1812,  Paris.     XIII  (1814),  Pt.  i,  1-371,  especially  218-314. 

3  J.  L.  Soret  et  Ed.  Sarasin:  Sur  la  polarisation  rotatoire  du  quartz.     Comptes  Rendus, 
XCV  (1882),  635-638. 

Idem:  Arch.  soc.  phys.  et  nat.  de  Geneve,  LIV  (1875)  253,  VIII  (1882),  5,  97,  201.* 

4  E.  Lommel:  Theorie  der  Drehung  der  Polarisationsebene.     Wiedem.  Ann.,  XIV  (1881), 
523-533. 

Idem:  Das  Gesetz  der  Rotationsdispersion.  Wiedem.  Ann.,  XX  (1883),  578-592,  in 
particular  592. 


ART.  78] 


ANISOTROPIC  MEDIA 


109 


in    which    a    and    ^20  are  constants  such  that  log  0=0.8555912   and  log 
/20  =  7.935i257  — 10.     The  result  is  in  thousandths  of  millimeters  (//). 

ROTATION  OF  THE  PLANE  OF  POLARIZATION  OF  QUARTZ 


Color 
extin- 
guished 

Fraun- 
hofer 
line 

Wave  length 
^in  ft  ft 
Angstrom 

Rotation 
per  mm. 
at  20°  C. 

Thickness 
necessary 
to  rotate 
90° 

Thickness 
necessary 
to  rotate 
1  80° 

Interference 
color  between 
crossed    nicols 

Red  

A 

760.4 

12.65° 

7.15 

114.30 

Green. 

Orange  
Yellow.... 

Green  
Blue  

a 
B 
C 
D-2 
D, 
E 
F 

718.36 
686.71 

656.  21 

589-  5J3 
588.912 
526.913 
486.074 

14-30 
15-75 
17-31 

21  .69 
21.725 

27-54 
32  .76 

6.  29 
5-7i 
5  i9 
4.14 
4.11 
3-26 

2  .75 

12.58 
11.42 
10.39 
8.29 
8.23 
6-53 
5-49 

Blue. 

Violet. 

Red. 
Orange. 

Indigo  
Violet  

G 

h 
H 

430.725 
410.12 
396.81 

42-59 

47-49 
5I.I9 

2.  IO 
1.89 

i-75   • 

4.  20 
3-79 
3-51 

Brownish  yellow. 
Yellow. 

The  effect  of  this  power  of  rotation  upon  the  interference  colors  may  be 
seen  from  Fig.  178,  which  represents  the  directions  of  vibration  of  the  red  A, 
orange  C,  yellow  D\,  green  E,  and  blue  F  rays  through  a  section  of  quartz 
8.23  mm.  in  thickness,  or  just  thick  enough 
to  rotate  the  plane  of  polarization  for  yellow 
DI  through  1 80°.  The  consequence  is,  when 
white  light  is  used,  that  the  yellow,  vibrat- 
ing at  right  angles  to  the  plane  of  the  ana- 
lyzer, is  extinguished,  and  the  plate  shows  a 
color  composed  of  the  resultants  of  the  re- 
maining rays,  namely,  the  sensitive  violet. 
If  the  analyzer  is  rotated,  each  color  in  suc- 
cession, as  it  becomes  perpendicular  to  the 
vibration  plane,  is  extinguished,  and  a  differ- 
ent interference  color  appears.  If  the  thick- 
ness of  the  section  had  been  but  4.12  mm., 
the  amount  of  rotation  of  the  DI  line  would 
have  been  but  90°,  consequently  the  sensitive  violet  would  have  appeared 
when  the  nicols  were  in  parallel  position. 

The  direction  of  rotation  of  the  plane  of  polarization  is  to  the  right  in 
some  crystals  and  to  the  left  in  others,  as  might  be  expected  from  their  enan- 


FIG.  178. — Diagram   showing 


110  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  79 

tiomorphous  forms.1  The  former  are  called  dextrogyrate  or  right  handed, 
the  latter  levogyrate  or  left  handed,  and  these  optical  directions  are  the  same 
as  the  crystallographic  rotation  directions.  Crystals  left  handed  optically 
are  left  handed  crystallographically,  and  vice  versa.  The  rotation  directions 
may  be  readily  recognized,  under  the  microscope,  by  noticing  that  when  the 
analyzer  is  turned  to  the  right  (clock-wise)  the  colors  pass  from  green  through 
blue,  purple,  and  red  to  green,  while  in  a  left-handed  quartz  the  reverse  order 
appears.2 

79.  Summary  of  Optical  Principles. — We  have  learned  that  light  consists 
of  vibratory  motion  of  the  ether,  and  that  it  is  transmitted  by  means  of  vi- 
brations taking  place  at  right  angles  to  the  direction  of  transmission  (Art.  18). 
If  we  consider  a  single  particle  in  motion,  its  movements  of  acceleration  and 
retardation  are  comparable  with  the  motion  of  a  pendulum,  or  the  projection 
of  the  motion  of  a  particle  around  a  circle.  When  a  particle  receives  two 
impulses  the  resulting  motion  will  depend  upon  the  direction,  amount,  and 
phase  of  the  components  (Arts.  27-28). 

There  are  two  kinds  of  media  in  which  light  may  travel;  isotropic,  in 
which  the  ease  of  vibration  is  the  same  in  all  directions,  and  anisotropic,  in 
which  the  ease  of  vibration  differs  in  different  directions  (Art.  29).  To  the 
former  class  belong  amorphous  substances  and  unstrained  crystals  of  the  iso- 
metric system;  to  the  latter,  crystals  of  the  tetragonal,  hexagonal,  orthorhom- 
bic,  monoclinic,  and  triclinic  systems. 

The  intensity  of  light  depends  upon  the  amplitude  of  the  vibrations,  the 
color  upon  the  rate  of  oscillation  (Arts.  31-32). 

The  wave  front  of  light,  in  isotropic  substances,  lies  at  right  angles  to  the 
direction  of  transmission  of  the  rays  (Art.  34).  When  a  ray  meets  with  an 
obstacle  to  its  free  movement,  it  is  partially  reflected  and  partially  refracted. 
The  angle  formed  by  the  reflected  ray  with  the  normal  to  the  reflecting 
surface  is  equal  to  the  angle  formed  by  the  incident  ray  (Art.  35),  but  the 
relation  between  the  incident  and  refracted  rays  is  such  that  the  ratio  of  the 
sine  of  the  angle  of  incidence  to  the  sine  of  the  angle  of  refraction  has  a  con- 
stant value,  known  as  the  index  of  refraction  (Arts.  37-38).  The  index 
of  refraction  of  a  ray  is  inversely  proportional  to  its  velocity  (Art.  40). 
When  light  passes  from  a  rarer  to  a  denser  medium,  the  refracted  ray  is  bent 
toward  the  normal,  when  from  a  denser  to  a  rarer,  away  from  it  (Art.  39). 

At  some  angle  of  incidence,  constant  for  the  same  substances,  the  refracted 
ray  is  parallel  to  the  separating  surface.  This  angle  is  known  as  the  critical 

1  J.  F.  W.  Herschel:    On  the  rotation  impressed  by  plates  of  rock  crystal  on  the  planes  of 
polarization  of  the  rays  of  light,  as  connected  with  certain  peculiarities  in  its  crystallization. 
Read  April  17,  1820.    Trans.  Cambridge  Phil.  Soc.,  I  (1821),  43-52. 

Anon:  Abstract  of  Mr.  HerscheVs  experiments  on  circular  polarization.  Edinburgh 
Phil.  Jour.  IV  (1821),  371-373. 

Anon:  Mr.  HerscheVs  experiments  on  plagiedral  quartz.     Ibid.  VI  (1822)  379. 

2  For  further  references  to  rotary  polarization  see  the  bibliography  given  at  the  end  of 
this  chapter. 


ART.  79]  ANISOTROPIC  MEDIA  111 

angle,  and  is  the  angle  whose  sine  is  the  reciprocal  of  the  index  of  refraction 
(Art.  41). 

While  in  isotropic  media,  in  general,  light  vibrates  with  equal  ease  in  every 
direction,  it  has  been  found  that  after  reflection  and  refraction  the  rays  do  not 
so  vibrate  but  are  polarized  so  that  the  vibrations  of  the  reflected  ray  take 
place  at  right  angles  to  the  plane  containing  the  incident  and  reflected  rays, 
while  the  refracted  ray  vibrates  in  the  plane  of  the  incident  and  refracted  rays 
(Art.  42). 

It  has  been  found  that  in  anisotropic  media  the  directions  of  maximum 
and  minimum  ease  of  vibration  lie  at  right  angles  to  each  other,  and  at  right 
angles  to  these  two  is  a  third  axis  of  intermediate,  though  not  necessarily 
mean,  ease  of  vibration  (Art.  47).  In  tetragonal  and  hexagonal  crystals  the 
vibrations  taking  place  in  the  basal  plane  are  equal  in  every  direction. 
The  direction  of  crystallographic  c  is  the  direction  of  maximum  or  minimum 
ease  of  vibration.  In  orthorhombic,  monoclinic,  and  triclinic  crystals,  the 
ease  differs  in  different  directions. 

When  light  enters  an  anisotropic  crystal  it  is  broken  up  into  two  rays 
vibrating  at  right  angles  to  each  other.  One  ray  vibrates  perpendicular  to 
the  plane  of  the  incident  ray  and  the  direction  of  propagation,  and  is  called 
the  ordinary  ray,  and  one  vibrates  in  the  planes  of  the  incident  and  the  re- 
fracted rays,  and  is  called  the  extraordinary  ray.  The  index  of  refraction  of 
the  former  is  represented  by  the  letter  a>,  that  of  the  latter  by  6  (Art.  48). 

In  tetragonal  and  hexagonal  crystals  there  is  but  one  direction  in  which 
there  is  no  double  refraction,  a  direction  at  right  angles  to  the  plane  of  equal 
ease  of  vibration,  consequently  parallel  to  crystallographic  c.  These  crys- 
tals are  called  uniaxial,  and  may  be  divided  into  two  classes:  those  in  which 
the  c  axis  is  the  direction  of  maximum  ease  of  vibration,  called  negative  crystals, 
and  those  in  which  the  c  axis  is  the  direction  of  minimum  ease  of  vibration, 
called  positive  (Arts.  49—50).  In  the  former  a>  >  e  and  in  the  latter  a)  <  e. 

The  principal  optic  section  of  a  uniaxial  crystal  is  one  containing  the 
axes  of  greatest  and  least  ease  of  vibration,  consequently  any  section  con- 
taining crystallographic  c  is  a  principal  section  (Art.  50). 

The  direction  of  vibration  of  the  ordinary  ray  is  at  right  angles  to  the 
direction  of  transmission  and  also  at  right  angles  to  the  plane  of  the  incident 
and  refracted  rays;  the  direction  of  vibration  of  the  extraordinary  ray  is 
parallel  to  the  tangent  to  the  ellipse  of  ray  fronts,  and  lies  in  the  plane  of  the 
incident  and  refracted  rays  (Art.  54). 

The  indicatrix  is  an  ellipsoid  whose  axes  are  the  principal  indices  of  re- 
fraction of  any  crystal  (Art.  59). 

The  ray  surface  and  the  wave  surface  do  not  coincide  since  waves  of  light 
are  not  transmitted  in  the  direction  of  the  rays  except  when  parallel  to  the 
axes  of  the  indicatrix.  The  movement  of  a  wave  is  measured  by  the  normal 
to  the  tangent  at  the  end  of  a  ray  (Art.  55). 


112  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  79 

Crystals  belonging  to  the  orthorhombic,  monoclinic,  or  triclinic  systems 
have  two  directions  along  which  the  light  waves  advance  with  equal  velocities 
and,  from  analogy  with  uniaxial  crystals,  they  are  called  biaxial.  The  optic 
axes  are  of  two  kinds :  primary  axes  or  binormals,  and  secondary  axes  or  bira- 
dials.  The  two  differ  very  slightly  in  position  in  a  crystal,  and  when  optic 
axes  are  spoken  of,  the  primary  axes  are  usually  meant  (Arts.  66-67). 

The  optic  or  axial  angle  is  the  angle  between  the  optic  axes.  Its  true 
value  is  indicated  by  2  F,  and  its  apparent  angle  in  air  by  iE.  If  measured  in 
oil,  etc.,  it  is  shown  by  2#(Art.  70). 

The  maximum  ease  of  vibration  in  biaxial  crystals  is  indicated  by  a,  the 
minimum  ease  by  c,  and  the  intermediate  by  b  (a>  b>  c).  The  correspond- 
ing refractive  indices  are  shown  by  «,  /?,  and 7  (<*</3<7)  (Art.  62). 

The  plane  of  the  optic  axes,  in  biaxial  crystals,  is  always  the  plane 
containing  a  and  c  (Art.  73). 

A  bisectrix  is  the  line  bisecting  the  optic  angle.  In  the  acute  angle  it  is 
called  the  acute  bisectrix;  in  the  obtuse  angle,  the  obtuse  bisectrix.  The  bi- 
sectrices always  coincide  with  a  and  c  (Art.  74). 

When  c  is  the  acute  bisectrix,  the  mineral  is  considered  positive;  when  a 
is  the  acute  bisectrix,  it  is  negative  (Art.  75). 

BIBLIOGRAPHY,  ROTARY  POLARIZATION 

1822.  A.  Fresnel:  Extrait  d'un  memoir e  sur  la  double  refraction  particuliere  que  presente  le 

cristal  de  roche  dans  la  direction  de  son  axe.     Ann.  chim.  et  phys.,  XXVIII  (1825), 

147-161. 
Idem:   Auszug  aus  einer  Abhandlung  Uber  die  eigenthumliche  Doppelbrechung  welch? 

der  Bergkry stall  in  Richtung  seiner  A  xe  darbietct.     Pogg.  Ann.  XXI  (1831),  2 76-290. 
1831.  G.  B.  Airy:  On  the  nature  of  the  light  in  the  two  rays  produced  by  the  double  refraction 

of  quartz.     Read  Feb.  21,  and  April  18,  1831.     Trans.  Cambridge  Phil.  Soc., 

IV  (1833),  79-1 23,  198-208. 
Idem:  Ueber  die  Natur  des  Lichts  in  den  beiden  durch  die  Doppelbrechung  des  Berg- 

krystalls  hervorgebrachten  Strahlen.     Pogg.  Ann.,  XXIII  (1831),  204-280. 
*837.  James  MacCullagh:  On  the  laws  of  the  double  refraction  of  quartz.     Read  Feb.  22, 

1836.     Trans.  Roy.  Irish  Acad.,  Dublin  XVII  (1837),  461-469. 
Idem:  Optical  laws  of  rock-crystal  (quartz],    Proc.  Irish  Acad.,  I  (1837-1840),  385- 

386. 
1852.  Broch:  Determination  du  pouvoir  rotatoire  du  quartz,  par  une  methode  applicable 

a  tous  les  phenomenes  chromatiques.     Ann.  chim.  et  phys.,  XXXIV  (1852),  119- 

124. 
1864.  J.  Stefan:  Ueber  die  Dispersion  des  Lichtes  durch  Drehung  der  Polarisaticnsebene  im 

Quartz.     Pogg.  Ann.,  CXXII  (1864),  631-634. 
Idem:  Same  title,  Sitzb.     Akad.  Wiss.  Wien,  L  (1865),  Abt.  II,  88-124. 

1869.  E.  Reusch:  Glimmer  combinationen.     Monatsber.  Akad.  Wiss.,   Berlin,   1869,  530- 

538. 

Idem:  Untersuching  iiber  Glimmer  combinationen.  '\  Pogg.  Ann.  CXXXVIII  (1869), 
628-638. 

1870.  E.  Verdet:  Leqons  d'optique  physique.     II,  Paris,   1870,   217-357.     Contains  very 

complete  bibliography,  to  1866,  of  thirteen  quarto  pages.* 


ART.  79]  ANISOTROPIC  MEDIA  113 

1877.  Viktor  v.  Lang:  Ueber  die  Drehung  der  Polar isationsebene  durch  den  Quartz.     Sitzb. 

Akad.  Wiss.  Wien,  LXXIV  (1877),  Abt.  II,  209-214. 

Idem:  Theorie  der  Circular  polarisation.     Pogg.  Erganzungsbd.  VIII  (1878),  608- 
624. 

1878.  J.  Joubert:  Snr  le  pouvoir  rotatoire  du  quartz  et  sa  variation  avec   la    temperture. 

Comptes  Rendus,  LXXXVII  (1878),  497-499. 

1882.  J.  Willard  Gibbs:   On  double   refraction   in   perfectly  transparent  media  which  ex- 

hibit the  phenomena  of  circular  polarization.     Amer.  Jour.  Sci.,  XXIII  (1882), 
460-476. 

1883.  W.  Voigt:  Theorie  des  Lichtes  fiir  vollkomtnen  durchsichtige  Media.     Wiedem.  Ann., 

XIX  (1883),  873-908,  especially  897-899. 
1885.  Gouy:  Sur  les  effects  simultanes  du  pouvoir  rotatoire  et  de  la  double  refraction.     Jour. 

d.  phys.,  IV  (1885),  140-159- 
1892.  D.  Goldhammer:  Theorie  electromagnetique  de  la  polarisation  rotatoire  naturelle  des 

corps  trans  parents.     Jour.  d.  phys.,  I  (1892),  205-209. 
1891.  Th.  Liebisch:  Physikalische  Krystallographie.    Leipzig,  1891,  502-519. 

1897.  Otto  Weder:  Die  Lichtbewegung  in  zweiaxigen  activen  Krystallen.     Neues  Jahrb., 

B.  B.,  XI  (1897-8),  1-45. 

1898.  H.  Landolt:  Das  optische  Drehungsvermogen.     Braunschweig,  2  Aufl.,  1898,  especi- 

ally 127-132. 
1901.  H.  C.  Pocklington:  On  rotatory  polarization  in  biaxial  crystals.     Phil.  Mag.,  II  (1901), 

361-370. 
1904.  H.  Dufet:  Recherches  experimentales  sur  Vexistence  de  la  polarisation  rotatoire  dans 

les  cristaux  biaxes.     Bull.  soc.  min.  France,  XXVII  (1904),  156-168. 
1906.  A.   Winkelmann:   Handbuch  der  Physik,   II,   article    Rotations  polarisation  by   P. 

Drude.     2  Aufl.,  1906,  1334-1364. 
191 1.  Paul    Kaemmerer:  Ueber  die  Interferenzerscheinungen  an  Flatten   optisch   aktiver, 

isotroper,    durchsichtiger    Kristalle   im    konvergenten  polarisierten   Licht.     Neues, 

Jahrb.,  1911  (II),  20-29. 


CHAPTER   VII 
LENSES 

80.  Definitions. — Of  primary  importance  in  a  microscope  are  the  lenses. 1 
A  lens  is  an  instrument  consisting  of  one  or  more  pieces  of  transparent 
material,  usually  glass,  and  having  curved  surfaces  which  may  be  spherical, 
elliptical,  parabolical,  or  cylindrical,  the  first  being  the  most  common.  There 
are  two  types  of  simple  lenses;  thin-edged,  convex,  or  converging  (Figs.  179- 
181),  and  thick-edged,  concave,  or  diverging  lenses  (Figs.  182-184).  If  the 


FIG.   179.     FIG.   180.     FIG.   181.       FIG.   182.     FIG.   183.     FIG.  184. 
FIGS.   179  TO  184. — Forms  of  lenses. 

centers  of  curvature  of  the  two  faces  are  on  opposite  sides  of  the  lens  and  the 
sum  of  the  lengths  of  the  two  radii  is  greater  than  the  distance  between  the 
centers,  the  lens  is  double-  or  bi-convex  (Fig.  179).  If  the  radius  on  one  side 
is  infinity,  the  lens  is  plano-convex  (Fig.  180).  If  the  centers  of  curvature 
are  on  opposite  sides,  and  the  sum  of  the  radii  is  less  than  the  distance  between 
the  centers,  the  lens  is  double-  or  bi-concave  (Fig.  182).  If  the  radius  on 
one  side  is  infinity,  the  lens  is  plano-concave  (Fig.  183).  If  the  radii  are 
both  on  the  same  side,  two  cases  may  occur,  depending  upon  the  relation 
between  the  radii  of  the  two  curves  and  their  centers.  Such  lenses  are 
meniscus  lenses,  either  converging  (Fig.  181)  or  diverging  (Fig.  184). 

81.  Axis,  Vertices,  and  Thickness  of  a  Lens. — The  axis  of  a  lens  (MM' , 
Fig.  185)  is  the  line  joining  the  centers  of  the  two  surfaces.     The  vertices 
(V,  V)  are  the  points  at  which  the  axis  intersects  the  surfaces.     The  thick- 
ness (VVr)  is  the  distance  between  the  vertices. 

82.  Optical  Center. — The  optical  center2  of  a  lens  is  such  a  point  that 
any  ray  passing  through  it  emerges  in  a  direction  parallel  to  that  in  which  it 

1  See  the  General  Bibliography  at  the  end  of  this  chapter. 

2  See  also  W.  F.  Durand:  A  practical  method  of  finding  the  optical  center  of  an  objec- 
tive and  its  focal  length.     Amer.  Mon.  Microsc.  Jour.,  VI  (1885),  141-145. 

114 


ART.  83] 


LENSES 


115 


entered.  In  Figs.  1 85  and  1 86  let  MP  and  M'P'  be  two  parallel  radii  of  a  bicon- 
vex and  a  biconcave  lens.  Then  evidently  //'  and  /"/'",  at  right  angles  to 
these  lines,  will  be  tangents  to  the  surfaces  of  the  lens.  Connect  P  and  P'. 
Obviously  a  ray  of  light  IP,  whose  direction  within  the  lens  is  PP',  will 
emerge  (P'R)  in  a  direction  parallel  to  IP  since  the  bounding  surfaces,  tt' 
',  are  parallel.  The  point  Oc,  where  the  line  PP'  cuts  the  line  MM', 


FIG.  185. 


FIG.    186. 


FIGS.  185  AND  186. — Optical  center  (Oc),  thickness  (VV).  and  centers  (M,  M')  of  biconvex  and  concavo- 
convex  lenses. 


is  the  optical  center  of  the  lens.  In  biconvex  and  biconcave  lenses  this  center 
falls  within  the  lens,  in  plano-convex  and  plano-concave,  it  is  at  the  vertex 
of  the  curved  surface,  and  in  a  concavo-convex  lens,  it  falls  outside. 

83.  Principal  Focal  Point. — The  principal  focal  point  of  a  convex  lens  is 
the  point  to  which  all  parallel  rays  sent  through  it  converge  (F,  Fig.  187).     It 


•L 


FlG.  187. — Real  and  virtual  focus  in  a  bi- 
convex lens. 


FIG.   188. — Virtual  focus  in  a  biconcave  lens. 


is  a  real  focus.  If  the  point  (F)  from  which  the  light  enters  the  lens  is  nearer 
than  the  principal  focal  point,  the  emerging  rays  will  diverge  and  appear  to 
come  from  a  point  on  the  same  side  of  the  lens  but  farther  from  it.  This  point 
(F'i)  is  called  the  virtual  focus. 

In  a  concave  lens  the  principal  focus  is  the  point  from  which  the  rays 
appear  to  diverge  (F,  Fig.  188).     It  is  a  virtual  focus. 


116 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  84 


84.  Conjugate  Foci  of  Convex  Lenses. — If  a  beam  of  light  emanates 
from  a  point  nearer  a  convex  lens  than  infinity,  the  rays  will  converge  at 
some  point  farther  from  the  lens  than  the  principal  focus.  Thus  a  beam  of 
light,  emanating  from  FI,  Fig.  189,  will  converge  at  F'\,  consequently  a  beam 
of  light  diverging  from  F\  will  converge  at  FI.  The  two  points  FI  and  F\ 
are  called  conjugate  foci. 


PIG.   189. — Conjugate  foci  of  a  biconvex  lens. 

85.  Refraction  through  Simple  Lenses. — A  lens  may  be  considered  as  having 
an  infinite  number  of  plane  faces,  represented  by  planes  tangent  at  every  point  of 
its  surface,  and  we  may  thus  determine  the  passage  of  light  through  it  by  means  of 
the  laws  of  refraction  in  isotropic  substances. 

In  order  that  our  formulae  may  be  applicable  to  all  kinds  of  lenses,  concave  as 
well  as  convex,  the  following  points  must  be  observed:  All  distances  measured  to 


r 


FIG.   190. — Refraction  at  a  curved  surface. 


the  left  of  the  lens  or  below  the  axis  are  written  with  a  minus  sign,  and  all  distances 
to  the  right  or  above,  with  a  plus  sign.  Also,  the  radius  of  curvature  of  the  face  of 
a  lens  is  considered  positive  if  its  center  is  to  the  right  of  the  vertex,  and  negative 
if  it  is  to  the  left. 

Let  the  glass  of  a  lens  first  be  considered  to  extend  indefinitely  to  the  right  from 
the  curve  A  V,  Fig.  190,  and  let  a  ray  of  light  enter  it  at  A ,  along  the  line  FiA ,  it' 
being  the  tangent,  and  AN  the  normal  to  it.  Let  AMV=  «,  FiAN  =  i  =  the  angle  of 
incidence, F\A  M  =  r  —  the  angle  of  refraction,  and  R\  =  A M  —  the  radius  of  curvature 
of  the  lens.  Since  in  any  triangle  the  sides  are  proportional  to  the  sines  of  the  oppo- 
site angles,  we  have,  in  the  triangle  MAF\M: 


(i) 


sn  a 


sn  « 


sin  a~FiA    "    —F\A 


ART.  85J 

and  in  the  triangle  MAF\M 


LENSES 


sin  r 


sin  r 


sin  r      MF'  ifi  — 


Dividing  (i)  by  (2)  we  obtain 


117 


(2) 


(3) 


sin  r       (f\-R1)-AFl 

in  which  n  =  the  index  of  refraction  of  the  material  of  the  lens,  usually  glass,  and 
RI  =  the  radius  of  curvature  of  the  lens. 

On  decreasing  the  angle  AFiM,  and,  consequently,  the  angle  of  incidence  /, 
the  minimum  value  will  be  reached  when  the  ray  coincides  with  the  axis  F\VF'\, 
and  JFi  will  equal  VFi,  and  AF\  will  equal  VF'i.  But  FF,=  -/1?  and  FF'i=/',. 
Substituting  these  values  in  equation  (3),  gives 


-fifi+f'iRi 

s 


Transposing,  we  have 


~7"T7T=  r, ~\.n—  i). 
Ji     J  i     Ri 

general  equation  of  conjugate  foci  in  two  media. 


(4) 


- 


FIG.    19 1. — Refraction  from  glass  through  a  curved  surface  into  air. 

In  a  manner  similar  to  the  above,  we  may  trace  the  passage  of  light  from  glass 
into  air,  and  we  have  (Fig.  191) 


i 

-fr 

J    2 


i 

„- 

/V2 


(5) 


where  w  is  the  index  of  refraction  of  the  substance  of  the  lens,/2  the  distance  of  the 
conjugate  focus  from  the  vertex  of  the  lens  and  lying  within  it,/'2  the  distance  of  the 
second  conjugate  focus  on  the  other  side  of  the  lens,  and  R2  the  curvature  of  the 
second  face  of  the  lens. 

The  effect  of  the  passage  of  light  through  a  lens  may  now  be  determined.  If  we 
consider  the  thickness  of  the  lens  as  infinitely  small,  /  (Fig.  192)  will  equal  zero,  and 
the  radii  of  curvature,  R±  and  R*,  will  meet  at  V. 

Adding  equations  (4)  and  (5),  we  have,  since  /'i=/2, 


(6) 


118 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  86 


This  is  the  general  equation  of  the  conjugate  foci  for  all  lenses,  disregarding  thickness, 
in  terms  of  the  radii  of  curvature  and  the  index  of  refraction  of  the  material  of  the  lens. 

If/!  =  oo  ,/'2  will  become  the  distance  (/)  of  the  principal  focus  (F)  from  the  center, 
and  equation  (6)  becomes. 


or 


f= 


(8) 


k-» 


— / -I 


PIG.   192. — Passage  of  light  through  a  lens. 

This  is  the  equation  of  the  principal  focus  of  any  lens  in  terms  of  the  curvatures  and  the 
index  of  refraction  of  the  lens,  thickness  being  disregarded. 
By  combining  (6)  and  (7)  we  have 


f 


(9) 


This  is  the  general  equation  of  the  principal  focus  for  all  lenses,  thickness  disregarded, 
in  terms  of  the  conjugate  foci. 

86.  Focus  of  Combined  Lenses. — In  Fig.  193  let  F\  be  the  point  where  the  paral- 
lel rays  /,  /',  and  I"  would  converge  after  passing  through  the  lens  A' A  A',  and  let 

/'i  represent  the  distance  of  this 

"'    B>  point,  the  principal  focus  of  the 

lens  A  A',  from  A.  If  now  a 
second  lens  B'BB'  is  inserted  at  a 
distance  of  AB  =  h  from  the  first 
lens,  and  in  the  path  of  the  rays 
coming  through  it,  the  light  fall- 
ing upon  the  second  lens,  being 
now  converging,  will  fall  at  F,  at 
a  distance  of  BF,  and  nearer  the 
second  lens  than  its  principal  focal 
point  F'2.  The  formula  for  com- 
bined lenses  may  be  obtained  from  this  diagram. 

The  rays  of  light  A'B'  pass  through  the  second  lens,  converge,  and  have  their 
virtual  focus  at  F,  which  is  the  real  focus  of  the  combination  as  well  as  the  con- 
jugate focus,  in  the  second  lens,  of  the  point  F\.  The  true  focal  distance 
of  the  combination  may  be  determined  by  equation  (9). 


PIG.   193. — Focus  of  combined  lenses. 


ART.  87J 


LJ-XSES 


119 


Now  the  distance  F\B=f\  —  h,  and  this  is  equal  to/i  —  h.  Substituting  f\  —  h 
for /i  in  equation  (9),  we  have 

7=~/T-A+/7  (lo) 

in  which/  =  the  principal  focal  distance  of  the  combination,/!  =  the  principal  focal 
distance  on  the  object  side  of  the  lens  A,  and/'2  =  the  principal  focal  distance  on  the 
image  side  of  the  lens  B. 

This  is  the  equation  of  the  principal  focus  of  combined  lenses,  thickness  disregarded, 
in  terms  of  the  principal  focus  of  each  lens. 

If  the  lenses  are  in  contact,  h  —  o,  and  equation  (10)  becomes 


7  "TTtjr: 

which  is  the  same  as  equation  (9),  as  it  should  be. 

87.  Gauss'  Method. — In  the  preceding  discussion  the  thickness  of  lenses 
was  disregarded.  If  this  is  introduced,  the  computations  are  much  more 
complicated,  though  the  problem  is  greatly  simplified  by  a  method  devised 
by  Gauss1  and  supplemented  by  Listing.2  The  method  is  applicable  to  all 


FIG.  194. — The  Gauss  points  of  a  simple 
lens. 


PIG.   195. — Location  of  nodal  points 
and  optic  center  of  a  lens. 


rays  which  make  a  small  angle  with  the  optic  axis  of  the  lens  combination. 
It  consists  of  reducing  a  lens  system  to  certain  location  points.  If  the  sys- 
tem is  well  centered,  it  may  be  reduced  to  three  pairs  of  location  or  cardinal 
points  along  the  axis,  called  the  focal,  nodal  and  principal  points,  and  by  these 
points  the  image  of  any  object  may  be  determined  for  any  system  of  lenses, 
at  least  with  approximate  accuracy.  The  essential  features  are  as  follows. 

Focal  Points  and  Planes. — The  focal  points  (Fi  and  F2,  Fig.  194)  are  the 
points  to  which  all  rays  parallel  to  the  axis  are  refracted,  or,  conversely, 

1  K.  F.  Gauss:  Dioptrische  Untersuchungen.     Gottingen,  1841. 

G.  Govi:  Rend.  R.  Accad.  Lincei,  IV(i888),  665-660.*     (Review  in  Jour.  Roy.  Microsc. 
Sec.,  1891,  122-126).     Gives  a  system  somewhat  different  from  that  of  Gauss. 

2  Johann    Benedikt    Listing:  Beitrag    aus    physiologischen    Optik.     Gottingen,    1845. 
Reprinted  in  Ostwald's  Klassiker  der  Exakten  Wissenschaften,  Nr.  147.    Leipzig,  1905. 


120 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  87 


all  rays  emanating  from  these  points  are  refracted  parallel  to  the  axis.  They 
are,  consequently,  the  principal  foci  of  the  lens.  FI  is  called  the  first  principal 
focal  point,  and  F2  the  second  principal  focal  point.  The  planes  through  these 
points,  and  at  right  angles  to  the  axis  of  the  lens  system,  are  called  the  focal 
planes. 

Principal  Points  and  Planes. — Suppose  a  ray  Fib' ',  Fig.  194,  emanating 
from  FI,  falls  upon  the  lens.  It  will  emerge  parallel  to  the  axis  along  the  line 
a'l'.  Another  ray,  la,  parallel  to  the  axis,  will  be  refracted  along  &F2.  The 
image  of  an  object  at  c',  when  viewed  from  FI,  appears  at  c,  and  an  object  at 
c,  when  viewed  from  F2,  appears  at  c'.  So  with  all  points  on  the  lines  cP\  and 
c'P2,  at  right  angles  to  the  axis  of  the  lens.  The  intersections  of  these  lines 
with  the  axis  at  PI  and  P2  are  called  the  principal  points,  and  the  planes 
through  these  points  and  perpendicular  to  the  axis,  the  principal  or  Gauss 
planes. 


FlG.  196. — Focal,  principal,  and  nodal  points  In  a  lens  system,  and  the  application  of  these  points  to 

the  location  of  a  refracted  image. 

Nodal  Points.1 — In,  Fig.  195  let  the  incident  ray  la,  and  the  refracted 
ray  a'R,  be  parallel.  The  points  where  the  two  rays  extended  cut  the  axis 
are  called  the  nodal  points  (N\  and  7V2). 

Focal  Distance. — The  distances  between  the  focal  points  and  the  principal 
points  are  called  the  principal  focal  distances.  FiPi=/i  (Fig.  194)  is  the  first 
principal  focal  distance,  and  P2F2=//2,  the  second.  They  are  true  focal 
distances. 

The  points  FI,  PI,  P2,  Ni,  7V2,  and  F2  all  lie  along  the  axis  of  the  lens 
system  and  bear  definite  relations  to  each  other.  Thus  the  distance  between 
the  first  focal  point  and  the  first  nodal  point  is  equal  to  the  distance  between 
the  second  principal  point  and  the  second  focal  point,  and  the  first  principal 
focal  distance  is  equal  to  the  distance  between  the  second  nodal  point  and  the 
second  focal  point.  Consequently  FiNi^P^Fz  and  PiP2  =  7Vi7V2,  Fig.  196, 
and/'2— /i=P2#2orPi#i.  Also/!  :  n'=f'2  :  n"  (nr  and  n"  being  the  indices 
of  refraction  of  the  media  on  either  side  of  the  lens).  If  the  media  on  either 
side  are  the  same,  n'  =  n"  and/i  =/72.  That  is,  the  two  principal  focal  distances 
are  the  same  when  the  media  on  either  side  of  the  lens  are  the  same,  and  further- 
more, since/i  =/'2,  the  principal  points  and  the  nodal  points  coincide. 

1  Knotenpunkte,  introduced  by  Listing.     Op.  cit. 


ART. 


LENSES 


121 


88.  Application  of  Gauss*  Cardinal  Points  to  the  Determination  of  the 
Image  Formed  by  a  Lens. — Let  it  be  required  to  find  the  position  of  the 
image  of  the  arrow  produced  by  the  lens  system  whose  cardinal  points  are 
shown  in  Fig.  196.     The  ray  A  a,  parallel  to  the  axis,  will  be  refracted  through 
the  focal  point  F2  to  some  point  on  the  line  aFzA'.     The  ray  AN\,  through 
the  first  nodal  point,  will  be  refracted  along  the  line  N2Af,  parallel  to  AN\. 
Where  the  two  lines  intersect  at  A'  is  the  required  point  of  the  image. 

In  the  same  manner  the  point  B  has  its  image  in  B',  at  the  intersection 
of  the  lines  bF2  and  N2B'. 

EQUATIONS  FOR  THE  DETERMINATION  OF  THE  CARDINAL  POINTS  OF  ANY  LENS 

SYSTEM 

SIMPLE  LENS 

89.  Lateral  Magnification. — Let  OOi,  Fig.  197,  be  an  object  at  a  distance  of 
—x  from  the  principal  focus  F,  which  itself  is  at  a  distance  of  — /  from  P,  the  inter- 
section of  the  axis  of  the  system  with  the  lens  P'P",  which  here  is  assumed  to  have 
no  thickness. 


To  determine  the  image  point  of  Oi,  two  rays  are  passed  through  it.  Rays 
parallel  to  the  axis  and  coming  from  the  left  have  their  focus  at  F',  consequently 
a  parallel  ray  through  Oi  will  cut  the  plane  P'P"  at  P"  and  be  projected  along  the 
line  P"F'  toward  O'\.  A  second  ray  through  Oi,  passing  through  F,  will  reach  the 
lens  at  P'  and,  since  it  passes  through  the  principal  focus,  will  be  projected  along 
P'O'i,  parallel  to  the  axis.  The  intersection  of  the  two  rays  at  O'\  is  the  position  of 
the  image  O\. 

Let  x'  represent  the  distance  of  this  image  to  the  right  of  F',  which  itself  is  at  a 
distance  of  /'  from  P. 

From  the  similar  triangles  PP'F  and  OO\F  we  have 


-  = 

-/      -x  y      -x 

and  from  the  similar  triangles  P"PF'  and  0'O\F' 

y      -y'        -y'     x' 

•- 


(n) 


(12) 


122 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  90 


The  ratio  of  the  size  of  the  image  (y'}  to  the  size  of  the  object  (y)  is  called  the 
lateral  magnification  of  a  lens  and  is  represented  by  /3.  From  equations  (n)  and 
(12)  we  have 

P  =  -^=~xJ'  (J3) 

or  -ff'=-xx'.  (14) 

90.  Convergence  of  a  Lens. — From  any  two  conjugate  foci,  as  0  and  O'  (Fig. 
197),  draw  rays  OP"  and  O'P",  making  angles  of  <f>  and  <p  with  the  axis,  and  forming 
the  angular  aperture  for  the  object  and  the  angular  aperture  for  the  image.  The 
convergence  (y)  of  the  lens  is  expressed  by  the  ratio  of  the  tangents  of  these  angles. 

tan  <p' 

(15) 


7  = 


tan  <p 


or 


/'+*'       -/-* 

v-    —     -J+? 

-/-* 

Adding  fxr  to  each  member  of  equation  (14)  we  have 

-/f  +/*'=-**'+/*', 


or 


Substituting  in  equation  (16)  we  have 


7  = 


(16) 


(i?) 
(18) 


(19) 


which  is  the  equation  of  the  convergence  of  a  lens. 


91.  Formation  of  Images  by  Lenses. — Applying  these  formulae  to  the 
determination  of  the  size  of  images  we  have,  from  the  equation  for  the  lateral 
magnification  of  a  biconvex  lens  (13),  disregarding  thickness  and  remember- 
ing that/  and  x  are  negative: 


FIG.  198. 


FIG.  199. 


-r    —(- 


-(-*) 


(13) 


If  the  object  is  at  a  distance  just  twice  the  focal  length  from  the  lens,  — /= 
—x  (Fig.  198.     Cf.  also  Fig.  197),  therefore 


=  i,  and  —yf  = 


ART.  92] 


LENSES 


123 


That  is,  when  the  object  is  twice  the  focal  distance  from  the  lens,  the  image  will 
be  the  same  size  as  the  object  and  will  be  real  and  inverted. 

If  the  object  is  at  a  distance  greater  than  twice  the  focal  length  of  the  lens, 
—  x>  —f  (Fig.  199),  therefore  equation  (13)  becomes 

~^-=^<i, and -/<>>. 
y       x 

That  is,  when  the  object  is  farther  from  the  lens  than  twice  its  focal  length,  the 
image  will  be  smaller  than  the  object  and  will  be  real  and  inverted. 

If  the  object  is  at  a  distance  greater  than  the  focal  length  but  less  than  two 
times  that  distance,  —  x  <  —f  (Fig.  200)  and 

f-i 


,  and  - 


That  is,  when  the  object  is  at  a  distance  greater  than  the  focal  length  but  less  than 


FIG.  200. 


FIG.  201. 


twice  this  distance  from  the  lens,  the  image  will  be  larger  than  the  object  and  will 
be  real  and  inverted. 

If  the  object  is  at  a  distance  less  than  the  focal  distance  from  the  lens, 
-\-x<  —f  (Fig.  201)  and 

-/-/\       y'      f 


That  is,  when  the  object  is  nearer  the  lens  than  its  focal  length,  the  image  will 
be  larger  than  the  object  and  will  be  virtual  and  erect. 

92.  System  of  Two  Faces. — Let  Fig.  202  represent  a  lens  of  two  faces.  The 
computation  may  be  simplified  if  we  consider  the  two  faces  as  independent  systems, 
the  light  first  passing  from  air  into  the  lens,  reaching  its  focus,  and  then  passing 
beyond,  through  the  lens,  to  the  second  surface  and  back  into  air.  The  two  faces 
will  be  treated  as  independent  systems,  and  will  be  spoken  of  as  the  first  and  second 
systems. 

In  Fig.  202  let 

FI   =  the  principal  focus  on  the  object  side  of  the  first  system, 
/i     =  the  focal  distance  of  FI, 

F'\  =  the  principal  focus  on  the  image  side  of  the  first  system, 
/'i    =  the  focal  distance  of  F'i, 
F2   =  the  principal  focus  on  the  object  side  of  the  second  system, 


124 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  92 


/2     =  the  focal  distance  of  F%  from  the  vertex  of  the  second  system, 

F'z  =  the  principal  focus  on  the  image  side  of  the  second  system, 

y'2    =  the  focal  distance  of  F'%  from  the  vertex  of  the  second  system, 

F     =  the  principal  focus  on  the  object  side  of  the  compound  system, 

/      =  the  principal  focal  distance  from  the  principal  plane  on  the  object  side, 

F'  =  the  principal  focus  on  the  image  side  of  the  compound  system, 

/'   =  the  principal  focal  distance  from  the  principal  plane,  on  the  image  side  of  the 

compound  system. 

It  will  be  seen  that  a  parallel  ray  of  light,  passing  from  left  to  right,  will  have  its 
principal  focus  on  the  image  side  of  the  first  system  at  F'I.  The  ray  will  now  pass 
on  into  the  second  system,  no  longer  as  parallel  light,  but  converging.  While  F'2 
is  the  focus  of  rays  entering  the  second  system  parallel  to  the  axis,  being  the  principal 


FIG.  202. — Passage  of  light  through  a  system  of  two  faces. 

focus  on  the  image  side  of  the  second  system,  it  will  not  be  the  focus  of  the  ray 
OPiF'i  which  is  now  not  parallel  to  the  axis,  although  it  was  so  originally.  Instead 
of  being  refracted  to  F'2  by  the  second  system,  therefore,  the  ray  will  be  refracted 
to  F',  a  point  which  is  conjugate  with  F'\  in  the  second  system.  That  is,  the  image- 
side  principal  focus  of  a  compound  system  of  lenses  is  at  a  point  which,  in  the  second 
system,  is  the  conjugate  of  the  image-side  principal  focus  of  the  first  system. 
From  equation  (14)  we  have,  consequently,  for  the  two  systems 


or 

and 

or 


-*.- 


• 


(20) 
(21) 
(22) 
(23) 


If  we  represent  by  2  the  distance  between  adjacent  focal  planes,  we  will  have, 
in  the  first  system,  2  =  the  distance  between  F'i  andF2  =  #'i,  whereby,  in  the  com- 
pound system  shown  in  Fig.  202,  equation  (21)  will  become 

_x==Ki. 

1         2 


(24) 


This  is  the  equation  of  the  distance,  on  the  object  side,  between  the  principal  focus  of  the 
combined  system  and  the  principal  focus  of  the  first  system. 


ART.  92]  LENSES  125 

In  like  manner,  equation  (23)  becomes,  in  the  second  system, 


*'2 

This  is  the  equation  of  the  distance,  on  the  image  side,  between  the  principal  focus  of 
the  second  system  and  the  principal  focus  of  the  combined  system. 
In  Fig.  197  let  6=PFPr, 


Likewise 


-/=tn  (26) 


tan 


These  are  Gauss'  equations  for  the  focal  lengths  (f  and  /')  of  a  compound  system. 
In  the  compound  system,  Fig.  202, 

tan#'i  =  7>  =  tan  #2  (of  the  second  system).  (28) 

Also  from  (15)  and  (19)  we  have 

_tan  0'2     ~/2 
T~tan  02  ~~x^" 
but  from  equation  (14)  we  have 

2,         ^/2  ^/2 

Since  F2F'2  =  a;2=^,  we  have 

tan  0'%     —  #2     —2 
tan  02  ~~  f'z  ~~  f'z 

V 

and  tan  O'z=  —  — ,  (tan  02).  (29) 

02  of  the  second  system  corresponds  to  0'i  of  the  first  (Fig.  202),  and  0'2  to  0' 
(Fig.  197),  therefore 

tan  0'=- ."/(tan  0'i).  (30) 

But  /'= ^>  (by  equation  27),  and  Ti  = ^->    therefore 

4-o-p   fl'   v    ^       •  9/9  j    •*•        f~5) ri   w 

tan  e'  =?-„  andy=/'i  (tan  0'i)« 

Substitute  these  values  in  equation  (30), 

(/' i  tan 


r-          y*         _          an  .  , 

-T  tan  0'i~  ^tan  0\  21  ' 

A  ray  coming  from  the  image  side  would  give 

f=fl£-  (32) 

These  values  of  f  and  f  are  -the  values  of  the  principal  focal  distances  of  the  compound 
system  in  terms  of  the  values  of  the  conjugate  foci. 


126 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  92 


By  comparison  with  the  refraction  through  a  simple  lens  (equation  5),  we  know 
that  if  the  rays  of  light  fall  upon  a  curved  surface  and  emerge  parallel  to  the  axis, 
f'2  =  oo,  whereby  equation  (5)  becomes 


1       n 


and 

H/2=    «, 


n  —  i 


(33) 


(34) 


The  general  equation  of  the  first  system  is  similar  to  (5),  and  we  have,  similarly, 
i  corresponding  to/2  of  the  second  system: 


_  »Ri  ^ 

n—i 

f-    Rl 

fl~     n—L 


(35) 
(36) 


These  are  the  values  of  the  conjugate  focal  distances,  fi  andfi,  of  a  simple  lens  in  terms 
of  the  index  of  refraction  of  the  material  of  the  lens  and  its  radii  of  curvature. 


As  before,  let  F'iF2  (Fig.  203)  =  21,  then 


Substitute  for/2  and/'i  their  values  from  equations  (33)  and  (35) 

2=_«R*      «R,+ 

n-i       n-i 


(37) 
(38) 
(39) 


This  is  the  equation  for  the  thickness  of  any  lens. 


ART.  92]  LENSES  127 

Substitute  the  values  of/i,/2  and  I  from  equations  (36),  (33),  and  (37)  in  equa- 
tion (32),  and  we  have 

Ri       nRz 

_  n  —  i     n  —  i        __  nR\Rz  __  /     >. 

---~~----' 


n  —  i 


Tliis  is  the  equation  of  the  principal  focal  length  of  any  lens. 
Equation  (39)  may  be  changed  to  the  form 


i\      t(n-iY 

'  (42) 


If  the  lens  is  infinitely  thin,  /  =  o,  and  equation  (42)  becomes 


which  is  the  same  as  equation  (7),  as  it  should  be. 
Substituting  I  for  x'\  in  equation  (21)  we  have 

-*1=~/1//1- 
Substitute  values  from  equations  (35)  and  (36)  we  have 

—  x\=          r —77,     ,\2F*  (43) 


Now  VF  (Fig.  2os)=fi-xi. 

Substitute  in  this  equation  values  from  (43)  and  (36) 


,     . 
(44) 


This  is  the  equation  for  the  distance  of  the  focus  of  any  lens  from  the  vertex  on  the  object 

side. 

In  a  similar  manner  the  second  principal  focus  of  a  biconvex  lens  may  be  deter- 
mined.    From  Fig.  203  and  equation  (23) 


Substituting  values  as  before,  from  equations  (34)  and  (33), 

-Rz      nRz 


T/'E"_       ^2       ;?  —  i      n  —  i  __  —  Rz(n  —  .     . 

n-i  2  («-i)2- 

This  is  the  equation  for  the  distance  of  the  focus  of  any  lens  from  the  vertex  on  the  image 
side. 


128 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  92 


Let  the  distance  between  the  principal  focus  and  the  principal  plane  be  repre- 
sented by  /,  and  let  d  be  the  distance  between  the  vertex  of  the  lens  and  the  prin- 
cipal plane.  From  Fig.  204  we  have, 


Substituting  values  from  equations  (32)  and  (44),  we  have 


Substitute  values  from  equations  (42)  and  (39) 


FIG.  204. 


But  from  equation  (39) 


n—i 

Substituting  this  value  in  (47),  we  have 

d~TIj~~^ 

Likewise, 

d'  =  , - 


(46) 


(47) 


(48) 
(49) 


These  are  the  equations  for  the  distances  from  the  vertices  to  the  principal  planes. 
From  Fig.  204  we  have 


Substitute  values  from  equations  (48)  and  (49), 

rt    r>         A   i  ^1  tR%  ^(n~I. 


/[  (n- 


n-i 


ART.  93]  LENSES 

But  according  to  equation  (37) 


129 


n(Rt-Ri)-t(n-i) 


n  —  i 


therefore 


This  is  the  general  equation  for  the  distance  between  the  principal  planes  of  any  lens. 


FIG.  205. — Under-corrected  spherical  aberration 
in  a  lens. 


FIG.  206. — Over-corrected  spherical 
aberration. 


93.  Aberration. — The  location  of  the  principal  points  by  Gauss'  theory, 
as  has  been  pointed  out,  is  accurate  only  when  the  pencil  of  light  differs  but 
slightly  from  the  axis  of  the  system.  In  practice  it  has  been  found  that 
unconnected  lenses  give  images  which  are  poorly  denned,  blurred,  or  distorted, 
an  effect  which  is  spoken  of  as  the  aberration  of  lenses. 1 


FIG.  207. — Astigmatism  in  a  lens.     (After  Wright.) 

Aberration  is  of  two  kinds,  spherical  and  chromatic. 

Spherical  Aberration. — Parallel  rays  of  monochromatic  light,  falling  upon 
a  spherical  lens,  will  be  found  to  be  refracted  to  different  points  upon  the  axis. 
Thus  in  the  spherical  biconvex  lens  in  Fig.  205,  the  marginal  rays  are  refracted 
to  F'  while  rays  near  the  center  converge  at  F".  This  distance  (F'F")  is 

L  For  methods  for  determining  the  aberration  of  lenses  see  Reginald  S.  Clay:  Treatise 
on  Practical  Light.  London,  1911,  211-243,  381-383. 

For  methods  for  correcting  aberration  see  W.   Zschokke:  Anschanliche  Darstellimg 
der  Entstehung  und  Hebung  der  spharischen  und  astigmatischen  Bildfehler.     Deutsche  Mech. 
Zeitung,  1910,  81-87,  93-97. 
9 


130 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  93 


called  the  longitudinal  spherical  aberration,  and  the  diameter  of  the  smallest 
circle  of  confusion  (cc')  is  known  as  the  lateral  spherical  aberration. 

If  to  a  lens,  such  as  that  shown  in  Fig.  205,  there  is  joined  another  lens 
whose  marginal  rays  fall  exactly  the  same  distance  beyond  F"  as  they  fall 
within  it  in  the  first,  the  resultant  is  zero.  Lenses  in  which  the  focus  of  the 
marginal  rays  falls  within  the  focus  of  the  central  rays  are  said  to  be  under- 
corrected  (Fig.  205),  and  those  in  which  the  reverse  is  true,  over-corrected 
(Fig.  206).  Lenses  corrected  for  spherical  aberration  are  said  to  be  aplanatic 
(a,  privative;  irhavrj,  to  wander). 

Another  correction  which  must  be  made  in  lenses  is  for  astigmatism. 
A  ray  of  light  falling  obliquely  upon  a  lens  (Fig.  207)  will  not  come  to  a  sharp 
focus,  for  while  the  lens  is  symmetrical  to  the  horizontal  ray  it  is  not  so  to  the 
vertical.  The  marginal  rays  of  the  horizontal  beam  will  intersect  at  the  same 
distance  from  the  lens  no  matter  what  the  inclination  of  the  ray  may  be,  but 
there  will  be  an  increasing  difference  in  the  length  of  the  upper  and  lower  rays 


FIG.  208. 


FIG.  209. 


FIG.  210. 


FIGS.  208  TO  210. — Images  as  viewed  through  an  orthoscopic  lens  and  through  those  which  are  un- 

corrected. 

with  increasing  inclination.  As  a  consequence  there  will  be  a  displacement 
of  the  latter  and  the  two  will  not  come  to  a  focus  at  the  same  point. 

Lenses  corrected  for  astigmatism  are  said  to  be  anastigmatic  (avd,  through- 
out, <my/wa,  a  point)  or  stigmatic. 

Another  distortion  for  which  correction  must  be  made  is  the  distortion  of 
the  image  whereby  rectilinear  lines  appear  as  shown  in  Figs.  209  or  210. 
Lenses  corrected  for  this  distortion  are  said  to  be  orthoscopic  (o/o0os,  straight, 
o-KOTrtlv,  to  look). 

Chromatic  A  berration. — If  ordinary  white  light  be  used  instead  of  mono- 
chromatic, another  form  of  aberration  results.  Since  different  colored  rays 
have  different  indices  of  refraction,  images  of  different  colors  will  be  formed  at 
different  distances  from  an  uncorrected  lens.  For  example,  in  a  biconvex 
lens  (Fig.  211)  violet  light  is  more  refracted  than  red,  and  will,  consequently, 
come  to  a  focus  nearer  the  lens.  Such  a  lens  is  said  to  be  chromatically 
under-corrected.  A  lens  whose  violet  rays  converge  beyond  the  red  is 
chromatically  over-corrected.  If  chromatic  aberration  is  corrected,  the 
lens  is  achromatic  (a,  privative,  \p^p^,  color) .  When  a  lens  is  corrected  for  the 
chromatic  aberration  of  two  colors,  it  does  not  necessarily  follow  that  the 


ART.  95] 


LENSES 


131 


remaining  colors  are  also  corrected,  and  intermediate  colors  may  come  to  a 
focus  within  or  beyond  the  corrected  focus.  By  using  combinations  of  lenses 
made  of  different  glasses  or  minerals,  lenses  may  be  chromatically  corrected 
for  three  colors  and  spherically  corrected  for  two.  Such  systems  may  consist 
of  ten  or  twelve  separate  lenses.  They  have  been  called  by  Abbe,  apochro- 
matic  (aTro,  from;  and  XP^t™,  color). 


94.  Angular  Aperture.  —  The  angle  which  the  rays  of  light  passing  through 
the  extreme  edges  of  the  opening  of  a  lens  make  at  the  focal  point  is  called 
the  angular  aperture  (PFP,  Fig.   21  2).  l 

95.  Numerical  Aperture.  —  If,  instead  of  air,  the  lens  be  immersed  in  a 
fluid  of  a  different  refractive  index,  it  is  clearly  evident  that  the  value  of  F 


will  be  changed  according  to  the  ratio  n  = 


sin  i 
j 

sin  r 


or  sin  i  =  n  sin  r,  where  i  is 


FIG.  211. — Chromatic  aberration. 
Lens  under-corrected. 


FIG.  212. — Angular  aper- 
ture of  a  lens. 


FIG.  213. — Change  of  focus 
due  to  an  immersion  fluid.  F  = 
focus  in  air,  F'  =  focus  in  oil. 


the  angle  made  by  the  ray  in  air  with  the  axis  of  the  lens,  and  r  the  angle  made 
by  the  ray  in  the  refracting  medium  (Fig.  213).  If  we  substitute  u  for  half 
the  angular  aperture,  the  equation  becomes 


It  may  be  written 


sin  i  =  n  sin  u. 
N.A.=n  sin  u. 


To  this  value  the  name  numerical  aperture  was  given  by  Abbe,2  and  by  it 
the  apertures  of  lenses  are  commonly  designated.     In  a  dry  system,  where  the 

1  See  W.  Blackburn:  The  theory  of  aperture  in  the  microscope.     Northern  Microsc.,  II 
(1882),  325-334. 

2  E.  Abbe:  On  the  estimation  of  aperture  in  the  microscope.     Jour.  Roy.  Microsc.  Soc., 
X.  S.,  I  (1881),  388-423. 

Idem:  The  relation  of  aperture  and  power  in  the  microscope.  Ibidem,  II  (1882),  300-309, 
460-473,  III  (1883),  790-812. 

See  also  W.  Blackburn:  Op.  cit. 

Mr.  Shadbolt:  Further  remarks  in  the  apertures  of  microscope  objeclhes.  Jour.  Roy. 
Microsc.  Soc.,  X.  S.,  I  (1881),  376-380. 

Charles  Hockin:  On  the  estimation  of  aperture  in  the  microscope.  Ibidem,  IV  (1884), 
337-347. 


132 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  96 


lens  acts  through  air,  n  =  i  and 

N.A.  =sin  u. 
In  an  immersion  system 

N.A.=n  sin  u. 

where  n  is  the  index  of  refraction  of  the  medium  lying  between  the  cover- 
glass  of  the  object  and  the  lower  lens  of  the  objective. 

96.  TABLE  OF  NUMERICAL  APERTURES  FOR  VARIOUS  ANGULAR  APERTURES 

Formula,  N.A.  =  n  sin  u. 


Angular  aperture 

n  =  i  .  oo  (air) 

w  =  i-33  (water) 

n  =  i  .52  (cedar  oil) 

2U=IO° 

0.087 

o.  116 

0.132 

i5° 

0.131 

0.174 

0.199 

20° 

0.174 

0.231 

o.  264 

25° 

o.  216 

0.287 

0.328 

30° 

0.259 

0-344 

0-394 

3s: 

0.301 

0.400 

0.458 

40° 

0.342 

0-455 

0.520 

45o 

0-383 

0.509 

0.582 

50° 

0.423 

0-563 

0.643 

55° 

0.462 

0.614 

0.702 

60° 

0.500 

0.665 

o.  760 

65° 

o-537 

0.714 

0.816 

70° 

0-574 

0.763 

0.872 

75° 

0.609 

0.810 

0.926 

80° 

0-643 

0-855 

0.977 

85° 

0.676 

0.900 

.028 

90° 

0.707 

0.940 

•075 

95° 

0-737 

0.976 

.  I2O 

100° 

0.766 

i  .019 

.164 

105° 

0-793 

1-055 

.205 

110° 

0.819 

.089 

-245 

H5° 

0.843 

.  121 

.281 

120° 

0.866 

.152 

-316 

I25o° 

0.887 

.ISO 

.348 

130 

0.906 

.205 

•378 

135° 

0.924 

.229 

.404 

I40° 

0.940 

•252 

.429 

As  the  angular  aperture,  and  consequently  the  amount  of  light  entering 
the  lens,  increases,  so  also  does  the  numerical  aperture,  and  as  a  result, 
other  things  being  equal,  the  lens  having  the  largest  angular  aperture  is  the 
most  desirable. 

97«  Apertometer. — In  practice  the  numerical  aperture  of  an  objective 
may  be  measured  by  means  of  an  instrument  called  an  apertometer,  devised 
by  Abbe1  in  1876.  The  instrument  consists  of  a  semi-circular  glass  plate  to 

1  Carl  Zeiss:  Description  of  Professor  Abbe's  aperlometer.  -Jour.  Roy.  Microsc.  Soc., 
I  (1878),  19-22. 

E.  Abbe:  Some  remarks  on  the  apertometer.     Ibidem,  III  (1880),  20-31. 

Ibidem:  Gesammelte  Abhandlungen,  I,  113-118,  227-243. 

E  M.  Nelson:  A  simplification  of  the  method  of  using  Professor  Abbe's  apertometer. 
Jour.  Roy.  Microsc.  Soc.,  1896,  592-594. 

See  also  Directions  for  using  the  Abbe  apertometer.     Zeiss'  circular,  Mikro.,  114. 


ART.  98]  f.l'.XSES  133 

which  are  attached  two  movable  black  metal  slides,  called  cursors  (b,  Fig. 
214),  which  may  be  seen  through  the  opening  a  by  reflection  from  the  beveled 
surface  below  it.  The  instrument  is  placed  on  the  stage  of  the  microscope  and 
the  edges  of  the  narrow  slit  at  a  are  brought  into  sharp  focus.  Upon  remov- 
ing the  eyepiece,  or  inserting  the  Bertrand  lens  or  an  auxiliary  objective  fur- 
nished with  the  instrument,  the  circular  edge  of  the  flat  glass  plate  will  appear, 
by  reflection,  as  though  it  were  directly  along  the  axis  of  the  microscope. 
The  cursors  are  now  moved  until  their  points  just  appear  at  the  edges  of  the 
field.  In  this  position  the  scales  are  read,  the  mean  of  the  two  values  indi- 
cated being  taken  as  the  angular  aperture  in  air.  Knowing  the  index  of 
refraction  of  the  apertometer  glass,  the  numerical  aperture  in  air  may  be 
calculated,  or  it  may  be  read  directly  from  a  second  scale  engraved  upon  the 
glass. 


FIG.  214.  —  The  Abbe  apertometer.     (Zeiss.) 

98.  Magnifying  Power.  —  The  amount  of  magnification1  of  a  lens  is 
the  ratio  of  the  size  of  the  image  to  the  size  of  the  object,  but  the  image  will 
appear  to  the  observer  to  be  at  a  distance  dependent  upon  the  eye  itself.  A 
normal  eye  can  see  a  small  object  most  clearly  when  it  is  ten  or  twelve  inches 
away.  This  distance  is  called  the  distance  of  most  distinct  vision  and  is  con- 
ventionally taken  at  250  mm.  (10  in.).  When  an  object  is  examined  through 
a  lens  by  a  normal  eye,  in  order  to  see  clearly,  the  lens  must  be  so  placed,  that 
is  focussed,  with  reference  to  the  object,  that  the  virtual  image  produced  will 
not  tire  the  eye.  In  lenses  of  short  focal  distance  this  results  when  the  object 
is  practically  at  the  principal  focus.  The  image  will  appear  to  be  at  the  dis- 
tance of  distinct  vision  or  250  mm. 

In  Fig.  215  the  image  AB  subtends,  in  the  eye  E,  an  angle  a  whose  tangent 

AB 

If  the  object  ab  were  situated  at  the  same  distance,  it  would  subtend 


/  T> 

an  angle  a    (a'EB)  whose  tangent  =         -      The  number  of  diameters  which 


1  E.  Abbe:  Note  on  the  proper  definition  of  the  amplifying  power  of  a  lens  ar  lens-system. 
Jour.  Roy.  Microsc.  Soc.,  2  ser.,  IV  (1884),  348-351. 

E.  Giltay:  Remarks  on  Prof.  Abbe's  Note  on  the  proper  definition  of  the  amplifying  power 
of  a  lens  or  lens-system.  Ibidem,  V  (1885),  960-967. 

E.  M.  Nelson:  Virtual  images  and  initial  magnifying  power.     Ibidem,  1892,   180-185. 


134 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  98 


the  lens  magnifies  is  naturally  the  ratio 


AB 

tan  a      EB     AB 


But  in  the  similar 


tan  a      a'B     a'B 

EB 

triangles  ACB  and  aCb,  AB  :ab  =  CB  :Cb  =  CB  :F,  therefore  the  number  of 
diameters  magnified  (TV)  would  be 

N_AB  _AB_CB 
a'B      ab      F   ' 

When  the  angles  a  and  aCb  are  small  and  the  eye  is  placed  near  the  lens, 
CB  =  EB  =  2$o  mm.,  and  the  equation  becomes 

A7  =  25° 

"  F 


K f—  •— »r 

FIG.     215. — Magnifying  power  of  a  lens. 

That  is,  the  number  of  diameters  which  a  lens  of  short  focal  length  will  mag- 
nify is  equal,  practically,  to  the  number  of  times  its  focal  distance  is  contained 
in  250.  Thus  a  lens  of  i  inch  (25  mm.)  focus  will  conventionally  have  a 
magnification  of  10  diameters,  but  while  this  will  give  the  apparent  size  of 
the  magnified  object  to  a  normal  eye,  it  will  not  be  the  apparent  size  to  a 
person  who  is  short-sighted.  If,  to  him,  the  distance  of  distinct  vision  is  only 

5  in.  (125  mm.),  N  becomes  —p-  and  the  lens  of  one  inch  focus  will  give  an 

apparent  magnification  of  5  diameters.1 

Sometimes  the  magnifying  power  of  a  microscope  is  expressed  in  terms  of 
areas.  Thus  a  lens  increasing  the  size  of  an  object  to  ten  diameters  will 
magnify  its  area  100  times;  a  magnification  of  50  diameters  is  equal  to  2500 
times,  and  so  on. 

JM.  C.  Montigny:  Bull.  Acad.  Roy.  Belgique,  XLIX  (1880),  670-678.*  Review  of 
preceding.  Difference  in  the  appreciation  of  the  apparent  size  of  microscopical  images  by 
different  obseners.  Jour.  Roy.  Microsc.  Soc.,  N.  S.,  I  (1881),  829-930. 


ART.  98]  LENSES  135 

GENERAL  BIBLIOGRAPHY 

1889.  Silvanus  P.  Thompson:  Notes  on  geometrical  optics.     Phil.  Mag.,  XXVIII  (1889), 

232-248. 
1891.  Idem:  The  measurement  of  lenses.     Jour.   Roy.   Soc.   Arts,  XL   (1891-2),   22-39. 

Reprinted  in  full  in  Jour.  Roy.  Microsc.  Soc.,  1892,  109-135. 
1891.  George  Macloskie:  The  dioptrical  principles  of  the  microscope.     Microscope,   XI 

(1891),  200-215.     Abstract  in  Jour.  Roy.  Microsc.  Soc.,  1892,  135-137. 
1895.  Th.  Marsson:  Beitrage  zur  Ttteorie  und  Technik  des  Mikroskops.    Zeitschr.  f.  angew. 

Mikrosk.,  I  (1895),  33-35,  65-69. 
1895.  Alfred  Daniell:  A  text-book  of  the  principles  of  physics.     3d  ed.,  New  York,    1895, 

533-542. 

1901.  Thomas  Preston:  The  theory  of  light.    London,  3d  ed.,  1901,  103-6. 
1904.  O.  D.  Chwolson:  Lehrbuch  der  Physik,  II.     Translated  from  the  Russian  by  H. 

Pflaum.     Braunschweig,  1904. 

1904,  Siegfried  Czapski:  Theorie  der  optischen  Instrumente  nach  Abbe.     Breslau,  1893. 
1904.  Rosenbusch-Wulfing:  Mikroskopische  Physiographic,  I-i,  Stuttgart,  1904,  118-147. 
1906.  A.  Winklemann:  Handbuch  der  Physik,  VI,  Optik.    Leipzig,  2  Aufl.  1906. 

1906.  P.  Drude:  Lehrbuch  der  Optik.    Leipzig,  2te  Aufl.,  1906.     An  English  translation 

by  Mann  and  Millikan,  London,  1902. 

1907.  Sir  A.  E.  Wright:  Principles  of  Microscopy,  New  York,  1907. 

1907.  Duparc  et  Pearce:  Traite  de  technique  mineralogique  et  petrographique,  I.  Leipzig, 

1907,  97-121. 
1907.  Lummer-Pfaundler-Muller-Pouillet:  Lehrbuch  der  Physik,  II,   Pt.   I,   Optik,    lote 

Aufl.,  1907. 

1909.  Arthur  Schuster:  Theory  of  optics.    London,  2nd  ed.,  1909. 

1910.  S.  O.  Eppenstein:  Aberration.     Encyclopedia  Britannica,  nth  ed.,  I,  1910,  54-61. 

1911.  Otto  Henker:  Lens.     Ibidem,  XVI,  421-427. 
1911.  Idem:  Microscope.    Ibidem,  XVIII,  392-407. 

1911.  Fred  Eugene  Wright.     The  methods  of  petrographic-microscopic  research.     Carnegie 
Institution  Publication  No.  158.     Washington,  1911,  14-56. 


CHAPTER  VIII 
THE  MICROSCOPE 

SIMPLE  MICROSCOPE 

99.  Hand  Lenses. — A  simple  microscope  (MIK/OOS,  small;  and  o-KorreTv,  to 
view)  is  one  which  consists  of  but  a  single  lens  or  of  a  system  acting  as 
a  single  lens,  and  gives  a  virtual  and  erect  image  larger  than  the  object.  The 
simplest  form  is  that  of  a  perfect  sphere,  the  primitive  lens  being  a  hollow 
glass  globe  rilled  with  water.  In  a  spherical  lens  the  distortion  (spherical 
aberration)  produced  by  the  outer  parts  is  extremely  great.  To  overcome 
this,  Wollaston  inserted  a  diaphragm  between  two  hemispheres  of  glass,  thus 


FIG.  216.  FIG.  217.  FIG.  218.  FIG.  219. 

FIGS.    216   TO    219. — Various  forms  of  simple  lenses.     Fi<?.  216,  Wollaston;  Fig.  217,  Brewster;  Fig 
218,  Stanhope  (Brewster);  Fig.  219,  Coddington. 

cutting  off  the  greater  portion  of  the  distortion  (Fig.  216).  Brewster's  lens 
consists  of  a  sphere  with  part  of  the  equator  cut  away,  as  in  Fig.  217.  He 
also  invented  the  so-called  Stanhope  lens  (Fig.  218)  which  consists  of  a  cylin- 
der of  glass  whose  two  ends  are  spherical  surfaces,  the  lower  one  being  of 
greater  radius  of  curvature  than  the  upper.  Somewhat  similar,  but  con- 


FlG.    220.  FlG.     221.  FlG.    222. 

FIGS.  220  TO  222. — Various  forms  of  doublets  and  triplets.     Fig.  220,  Wollaston's  doublet; 
Fig.  221,  Achromatic  doublet;  Fig.  222,  Steinheil  triplet. 

sisting  of  a  cylinder  cut  from  a  sphere  and  having  a  groove  around  the  waist 
to  serve  as  a  diaphragm,  is  the  lens  devised  by  Coddington  (Fig.  219). 

More  successful  in  overcoming  aberration  are  lenses  made  of  various 
combinations  of  two  or  three  lenses,  and  called  doublets  or  triplets.     Wollas- 

136 


ART.  991 


THE  MICROSCOPE 


ton's  doublet  (Fig.  220)  consisted  of  two  plano-convex  lenses  of  different 
sizes  with  the  plane  surfaces  toward  the  object.     An  achromatic  doublet  is 


FIG.  223. — Lens  stands.     (Leitz.) 


shown  in  Fig.  221 ;  it  consists  of  one  flint-glass  and  one  crown-glass  lens.  The 
Steinheil  lens  consists  of  a  central  crown  glass  between  two  of  flint  (Fig.  222), 
and  gives  corrections  for  both  spherical  and  chromatic  aberration. 


FlG.  224. — Steinheil 
aplanatic  lens. 
(Leitz,) 


FlG.  225. — Coddington  magnifier 
with  handle  for  laboratory  use. 
(Bausch  and  Lomb.) 


FIG.  226. — Codding- 
ton magnifier  with  fold- 
ing case.  (Bausch  and 
Lomb.) 


Lenses  are  variously  mounted,  depending  upon  the  purposes  for  which 
they  are  intended.     Fig.  224  shows  a  lens  mounted  for  use  in  a  lens  holder 


FIG.  227. — Hastings  aplanatic  triplet. 
(Bausch  and  Lomb.) 


FIG.   228. — Steinheil  aplanatic  triplet. 
(Leitz.) 


such  as  is  shown  in  Fig.  223 ;  Fig.  225  has  a  handle  for  laboratory  use,  and  Figs. 
226  to  228  are  mounted  in  folding  cases  for  field  use. 


138  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  100 

There  are  several  points  to  be  considered  in  choosing  a  hand  lens  for  petro- 
graphic  work.  Two  of  the  most  important  are  flatness  of  field1  and  depth  of 
focus.2  Since  hand  specimens  of  rocks  have  surfaces  which  are  generally 
quite  rough,  it  is  desirable  that  the  lens  should  be  able  to  show  clearly  points 
which  are  in  slightly  different  planes.  A  focal  length  of  about  i  in.  (25  mm.) 
is  perhaps  the  best  for  petrographic  work,  since  it  is  short  enough  to  permit 
of  considerable  magnification  and  long  enough  to  allow  plenty  of  light  to  fall 
upon  the  specimen  even  when  one  is  examining  a  mineral  within  a  small 
depression.  The  Hastings  Aplanatic  Triplet  with  i  in.  focus  and  magnifying 
ten  diameters  (Fig.  227)  is  excellent  but  rather  expensive.  Leitz's  Aplanatic 
(Figs.  224  and  228)  is  recommended  as  a  moderate  priced  lens,  while  the 
Coddington  magnifier  (Figs.  225-226)  is  fairly  satisfactory,  and  cheap. 

In  regard  to  the  care  of  a  pocket  lens,  much  that  is  said3  in  regard  to  the 
care  of  lenses  in  general  will  apply.  Carried  in  the  pocket,  it  is  likely  to  be- 
come dusty,  and  care  should  be  taken  not  to  scratch  the  surface  in  cleaning  it. 
If  kept  in  a  small  purse  it  will  not  only  be  protected  but  will  be  prevented 
from  readily  slipping  out  of  the  pocket. 

In  using  a  hand  lens  it  is  to  be  remembered  that  the  nearer  it  is  held  to 
the  eye,  the  greater  will  be  the  field  of  view. 

COMPOUND  MICROSCOPE 

100.  Formation  of  the  Image. — The  compound  microscope4  usually  con- 
sists of  two  systems,  themselves  compound.     The  light  is  reflected  from  the 
mirror   (Sp,  Fig.  231)   and  passes  through  the  diaphragm  (CD,  Fig.  229) 
which  is  placed  at  the  point  where  the  rays  cross,  a  point  known  as  the  en- 
trance pupil  of  the  microscope.     It  then  passes  through  the  objective  or 
object-lens  and  produces  a  real,  enlarged  image  of  an  object  placed  at  O\. 
This  image  would  normally  fall  at  O2,  but  owing  to  the  passage  of  the  rays 
through  the  lower  or  field  lens  of  the  eyepiece  or  ocular,  they  converge,  and 
the  image  appears  at  O3;  that  is,  O%  and  O3  are  conjugate  foci  of  the  field 
lens.     The  eye  is  placed  above  the  upper  or  eye  lens  of  the  ocular  at  the 
point  EP  where  the  rays  cross,  a  point  known  as  the  Ramsden  disk,  pupil  of 
the  eyepiece,  or  exit  pupil.     The  image  03,  which  lies  in  the  focal  plane  (F2)  of 
the  eye  lens,  acts  as  the  object  for  this  lens  and  appears  as  a  virtual,  enlarged 
image  (O4)  at  the  distance  of  distinct  vision  (C)  250  mm.  from  the  eye. 

101.  Optical  and  Mechanical  Tube  Lengths. — The  plane  in  which  the 
rays  cross  at  FI,  Fig.  229,  is  known  as  the  posterior  focal  plane  of  the  objec- 

1  Art.  145,  infra. 

2  Art.  144,  infra. 

3  Art.  198,  infra. 

4  For  a  history  of  the  microscope  see  R.  J.  Petri:  Das  Mikroskop,  248  pp.,  191  figs. 
Also  E.  M.  Nelson:  Development  of  the  compound  microscope.     Trans.  Middlesex  Nat. 

Hist,  and  Sci.  Soc.,  1886-7,  103-111.*     Review  in  Jour.  Roy.  Microsc.  Soc.,  1888,  136-7. 


ART.  101] 


THE  MICROSCOPE 


139 


tive;  where  they  converge  at  F2,  the  anterior  focal  plane  of  the  ocular.  The 
distance  between  these  two  planes  is  called  the  optical  tube  length1  and  is 
expressed  by  A .  The  mechanical  tube  length  (L)  is  the  distance  between 
the  upper  end  of  the  body  tube  and  the  shoulder  of  the  objective  screw.  It 


FIG.  229. — Passage  of  light  through  a  compound  microscope. 


varies  somewhat  in  the  instruments  of  different  makers,  being  between  160 
and  170  mm.  in  German  microscopes  and  between  200  and  250  mm.  in  those 
of  English  makers.  A  number  of  years  ago  the  American  Society  of  Micros- 

1  See  Frank  Crisp:  On  optical  tube  length,  an  unconsidered  element  in  the  theory  of  the 
microscope.     Jour.  Roy.  Microsc.  Soc.,  2  ser.,  Ill  (1883),  816-820. 


140  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  102 

copists1  adopted  two  standards,  one  of  160  mm.  (6.3  in.)  for  short,  and  one  of 
216  mm.  (8.5  in.)  for  long  tubes.  There  seems  to  be  no  special  advantage  of 
one  over  the  other  except  that  the  short  tube  is  often  more  convenient. 
Objectives  made  for  a  certain  tube  length  should  be  used  for  that  length  and 
for  no  other,  if  the  best  results  of  which  it  is  capable  are  to  be  obtained.  Many 
instruments  are  so  arranged  that  the  tube  is  capable  of  adjustment  to  dif- 
ferent lengths. 

102.  Focal  Length.  —  The  focal  length  of  the  compound  microscope  is 
expressed  by  the  equation 

„    FiF, 

~ 


where  F\  equals  the  focal  length  of  the  objective,  F2  the  focal  length  of  the 
ocular,  and  A  the  optical  tube  length. 

103.  Magnifying  Power.  —  It  was  shown2  that  the  magnifying  power  of 
a  simple  microscope  is  expressed  by  the  equation 


The  same  equatioji  expresses  the  magnification  of  the  compound  microscope, 


same  equatioji  expresses  the  magn 
for 

_/_ 
tan  ai       F    ^250 

tan  a        y1  '_    =  F 
250 

where  a  =  half  the  angle  subtended  by  the  object  when  viewed  at  a  distance 
of  250  mm.  without  the  microscope,  and  «i  =  half  the  angle  formed  by  the 
image  seen  through  the  microscope.  That  is,  the  magnifying  power  of  a 
compound  microscope  is  the  ratio  of  the  angle  subtended  at  the  eye  by  the 
image  at  the  distance  of  distinct  vision,  to  that  subtended  by  the  object  at 
the  same  distance.  As  with  a  simple  lens,  the  actual  magnification  in  the 
compound  microscope  differs  for  different  observers,  depending  upon  the  . 
power  of  accommodation  of  the  eye. 

104.  Field  of  View.  —  The  field  of  view  decreases  with  the  magnifying 
power.  Roughly,  the  reciprocal  of  the  magnifying  power  in  diameters,  mul- 
tiplied by  five,  will  give,  in  fractions  of  an  inch,  the  size  of  the  field  when  a 
Huygens  eyepiece  is  used.  Thus  with  a  microscope  magnifying  10  diame- 

1  Report  of  the  committee  of  the  American  Society  of  Microscopists    on    uniformity  of 
tube  length.     Microscope,  X  (1890),  297. 

The  tube  lengths  used  by  various  makers  are  as  follows:  Bausch  &  Lomb,  Beck,  Reichert, 
and  Zeiss  160  mm.;  Fuess,  Leitz,  and  Seibert,  170  mm.;  Swift,  and  Nachet,  200  mm. 

2  See  also  Articles  98  and  149. 


ART.  105] 


THE  MICROSCOPE 


141 


ters,  an  object  1/2  in.  in  diameter  may  be  seen;  with  20  diameters,  an  object 
1/4  in.  in  diameter;  with  100  diameters  an  object  1/20  in.  in  diameter;  with 
=500  diameters,  an  object  i/ioo  in.  in  diameter,  and  so  on. 


THE  PETROGRAPHIC  MICROSCOPE 

105.  Description. — A   petrographic   microscope   is  much  more  compli- 
cated than  a  biologic  microscope,  for  although  the  magnifying  powers  used 


: 

FIG.  230.  FIG.   231. 

FIGS.   230  AND  231. —  Mineralogical  microscope.     Large  stand  AM.     (Leitz.) 

are  not  so  great,  yet  there  are  attached  to  it  appliances  for  special  examina- 
tions which  require  careful  adjustment. 

Essentially,  a  petrographic  microscope  consists  of  a  stand  having  a  heavy 
foot  (F,  Fig.  231)  and  an  upright  pillar  (Si)  to  which  are  attached  a  revolv- 
ing stage,  and  the  arm  or  limb  (OT)  carrying  an  adjustable  tube  (T).  Below 
the  stage  is  a  mirror  (Sp)  and  a  polarizer  (P).  The  tube  carries  the  ocular  or 
eyepiece  (HO),  the  objective  (O),  and  the  analyzer  (A). 


142  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  106 

The  Mechanical  Parts  of  a  Petrographic  Microscope 

1 06.  Foot. — The  foot  (F,  Fig.  231)  of  the  microscope,  in  American  and 
German  instruments,  is  usually  made  in  the  form  of  a  horseshoe ;  in  English 
instruments,  in  the  form  of  a  tripod  (Figs.  320-321).     When  in  the  form  of  a 
horseshoe,  the  intermediate  parts  should  be  so  cut  away  that  the  instrument 
rests  only  upon  three  points,  which  should  be  covered  with  leather  or  felt. 
Of  whatever  form,  it  should  be  of  sufficient  weight  to  counter-balance  the 
instrument  when  it  is  tilted  backward  as  far  as  possible. 

107.  Pillar  or  Post. — The  pillar  or  post  (Si}  is  the  upright  attached  to 
the  foot,  and  carries,  at  its  upper  end,  a  hinge  (G)  which  may  be  loosened  or 
tightened  by  means  of  a  bar  or  a  spanner.     The  hinge  should  work  smoothly 
and  yet  be  tight  enough  to  prevent  the  upper  part  of  the  instrument  from  run- 
ning backward  by  its  own  weight  and,  by  the  increased  momentum,  overturn- 
ing the  instrument.     It  is  safer  never  to  allow  the  microscope  to  remain  in 
a  tilted  position  when  not  in  use.     The  habit  of  always  placing  it  upright  when 
leaving  the  table  may  some  time  save  it  from  an  expensive  fall. 

108.  Limb  or  Arm. — The  limb  or  arm  (OT)  is  fastened  by  the  hinge  (G) 
below,  and  carries,  at  its  upper  end,  fine  (JE)   and  coarse  (gE)  adjustment 
screws  for  the  draw-tube,  and  at  the  lower  end  a  support  for  the  stage.     The 
arm  is  sometimes  made  with  an  unnecessary  handle,  which  only  makes  more 
weight  above  the  hinge. 

109.  Stages,  Simple  and  Mechanical. — The  stage  is  the  table,  usually 
circular,  upon  which  the  thin  section  is  placed  for  examination.  In  almost 
all  microscopes  it  is  so  arranged  that  it  may  be  rotated  and  the  amount  of 
rotation  determined  by  means  of  a  graduated  scale  and  vernier,  reading  to 
minutes.  In  its  simplest  form  (Fig.  307)  the  stage  is  a  circular  disk  with  an 
opening  in  the  center  through  which  the  light  passes.  The  thin  section  usu- 
ally is  held  in  place  by  spring  object  clips,  such  as  are  shown  in  the  figure. 
The  objection  to  such  clips  is  that  they  are  likely  to  catch  under  the  edges  of 
the  cover-glass,  and  either  break  or  strip  it  from  the  mount.  It  is  also  diffi- 
cult to  move  the  slide  a  very  small  distance,  such  as  may  be  necessary  in 
centering  a  mineral  for  use  with  a  high-power  objective.  One  is  likely  to 
move  the  slide  too  far,  first  in  one  direction  and  then  in  another.  To  over- 
come this,  various  types  of  mechanical  stages  have  been  devised. 

Among  mechanical  stages,  one  of  the  simplest  and  most  satisfactory  is 
that  designed  by  Hirschwald1  (Fig.  232).  This  stage  possesses  the  advantage 
of  holding  a  slide  firmly  in  place  with  no  interference  by  spring  clips.  It 
has  two  motions,  excellently  adapted  to  a  rapid  inspection  of  every  part  of  a 

1  J.  Hirschwald:  Ueber  ein  neues  Mikroskopmodell,  etc,  Centralbl.  f.  Min.,  etc.,  1904, 
629-630. 


ART.  109] 


THE  MICROSCOPE 


143 


thin  section,  one  produced  by  sliding  the  plate  S  within  the  grooves  over  the 
depressed  portion  A,  the  other  by  sliding  the  thin  section  itself  between 
the  bars  b  and  bi,  one  of  which  holds  the  slide  firmly  by  means  of  the  spring 
/.  The  entire  plate  S  may  be  removed  and  a  plain  plate  inserted,  making, 
thus,  a  flat  stage  with  which  the  usual  spring  clips  may  be  used.  Johannsen1 
suggested  several  improvements,  among  others  the  addition  of  graduations 


FIG.  232. — The  Hirschwald  stage.     (Fuess.) 

along  the  top  bar  and  sides  of  the  sliding  plate  whereby  any  desired  point  in 
a  thin  section  may  readily  be  found  again. 

Another  mechanical  stage2  is  shown  in  Fig.  233.  This  stage  carries, 
besides  the  usual  division  of  the  circumference  into  a  scale  reading  to  minutes 
by  means  of  verniers,  two  sliding  portions  controlled  by  the  screws  5  and  sf, 
one  of  which  has  micrometer  divisions  to  o.oi  mm.,  the  other  a  screw  with 
strong  pitch  which  serves  as  a  rapid  finder  with  low  magnifications.  Its 
movement  is  read  to  0.5  mm.  by  the  scale  on  the  stage.  By  means  of  these 
screws  it  is  possible  to  measure,  sufficiently  accurately,  by  a  much  more  rapid 
method  than  by  a  micrometer  ocular,  the  dimensions  and  proportions  of  the 
minerals  which  occur  in  a  slide.  By  their  means,  also,  it  is  possible  to  find 
any  mineral  once  determined,  the  position  of  the  slide  on  the  stage  being  fixed 
by  the  guide  strip  w. 

There  are  several  objections  to  this  stage:  its  thickness,  its  liability  to 

1  Albert  Johannsen:  Some  simple  improvements  for  a  petrographical  microscope.  Amer. 
Jour.  Sci.,  XXIX  (1910),  438. 

2R.  Fuess:  Ueber  Mikroskope  fiir  krystallographische  und  petrographische  Untersuch- 
ungen.  Neues  Jahrb.,  B.B.,  CII  (1891),  55-89,  in  particular  57-58. 

C.  Leiss:  Die  optischen  Instrumente  der  Firma  R.  Fuess,  Leipzig,  1889,  185. 


144 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  110 


lost  motion,  and  the  interference,  in  certain  positions,  of  the  micrometer 
screws  with  reading  the  stage  vernier. 

Another  type  of  mechanical  stage  is  that  shown  in  Figs.  230  and  231. 


FIG.  233. — Mechanical  stage. 
(Fuess.) 


1/2  natural  size. 


FIG.  234. — Attachable  mechanical  micrometer 
stage.     (Leitz.) 


It  possesses  the  advantage  of  being  thin  and  concealed  within  the  stage,  the 
amount  of  movement  being  read  through  openings  in  the  latter.  It  is  not, 
however,  adapted  to  fine  readings,  which  must  be  made  by  means  of  a  microm- 
eter ocular. 

An  attachable  mechan- 
ical stage  is  shown  in  Fig. 
234.  This  may  be  read  to 
tenths  of  millimeters  and 
is  detachable,  so  that  ordi- 
narily it  will  not  be  in  the 
way.  It  is  fastened  to  the 
stage  by  means  of  a  thumb 
screw. 

Somewhat  similar  is  the 
stage  shown  in  Fig.  235. 

no.  Verniers. — In  con- 
nection with  the  fine  ad- 
justment screw,  and  also 
attached  to  the  sides  of 
the  stage  of  most  micro- 
scopes, there  are  certain 

auxiliary  scales  called  verniers,  after  the  inventor,  Pierre  Vernier  of  Burgundy 
(1631).  These  verniers  may  be  of  various  kinds,  but  in  all  of  them,  the  divi- 
sions are  a  little  longer  or  a  little  shorter  than  the  divisions  of  the  main  scale. 


FIG.  235. — Attachable  mechanical  stage.     (Bausch  and  Lomb.) 


ART.   Ill] 


THE  MICROSCOPE 


145 


If,  for  example,  in  Fig.  236,  nine  parts  of  the  main  scale  are  equal  to  ten  parts 
of  the  vernier,  then  each  division  of  the  latter  is  one-tenth  of  a  space  shorter 
than  a  space  on  the  former.  As  the  vernier  is  made  to  coincide  with  succes- 
sive divisions  of  the  larger  scale,  each  successive  coincidence  indicates  an  ad- 
vance of  i/io  of  a  division.  In  Fig.  236  each  division  of  the  scale  is  o.i  in. 
therefore  each  successive  mark  on  the  vernier  indicates  a  movement  of  o.i 
of  o.i  or  o.oi  in. ;  and  in  a  scale  so  marked,  the  number  of  whatever  line  on  the 
vernier  coincides  with  some  line 
on  the  scale  will  indicate  the 
number  of  hundredths  of  an 
inch  of  movement. 

There  are  two  classes  of  ver- 
niers, direct  and  retrograde. 
Scales  such  as  that  just  de- 
scribed, in  which  the  spaces  on 
the  vernier  are  shorter  than  those 
on  the  main  scale  and  the  num- 
bering and  the  lines  successively 
coinciding  are  in  a  forward  direc- 
tion, are  called  direct.  A  retro- 
grade vernier  has  spaces  larger 
than  those  on  the  main  scale 
(Fig.  237),  and  the  numbering  and  the  successive  coincidences  are  in  the 
reverse  direction.  In  both  kinds,  the  difference  between  a  division  on  the 
vernier  and  the  scale  is  called  the  least  count,  and  is  the  measure  of  its 
smallest  reading. 

The  verniers  on  circular  scales  usually  read,  not  to  tenths,  but  to  minutes 
or  seconds.  In  Fig.  238  the  least  count  is  one  minute.  The  main  scale  is 
divided  to  half  degrees  and  29  of  its  parts  coincide  with  30  of  the  vernier. 

In  verniers  where  half  divisions  are 
the  basis  of  subdivision,  one  must  use 
great  care  not  to  neglect  a  part  of  the 
reading,  such  as  the  half  degree  in  Fig. 
238,  which  reads  o°  33'.  If  one  is  in 
doubt  whether  a  division  of  the  vernier 
coincides  with  a  division  on  the  scale, 

one  should  notice  that  those  on  either  side  differ  an  equal  amount.     Where 
several  divisions  seem  to  coincide,  read  the  middle  one. 


FIG.  236. — A  direct 
vernier. 


FIG.  237. — A  retro- 
grade   vernier. 


20  10 

FIG.  238. — Vernier  on  circular  scale. 


in.  Body  Tube.— The  body  tube  (T,  Fig.  231)  is  the  hollow  cylinder  to 
which  most  of  the  optical  parts  are  attached.  It  sometimes  consists  of  but  a 
single  tube  (Fig.  311),  but  generally  it  carried  within  it  a  second  cylinder 
called  the  draw  tube  (TA,  Fig.  231).  The  latter  has  engraved  upon  its  side 


146  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  112 

a  millimeter  scale  which  shows  the  mechanical  tube  length.  The  draw  tube, 
in  some  microscopes  (Figs.  307  and  318),  may  be  extended  by  simply  pulling  it 
out,  and  it  is  held  in  place  by  friction,  but  in  the  more  expensive  instruments  it 
is  moved  by  means  of  a  rack  and  pinion  (OcE,  Fig.  231). 

The  methods  of  attaching  the  optical  parts  to  the  tube  are  quite  different 
in  different  instruments,  and  will  be  described  in  greater  detail  in  the  section 
dealing  with  the  various  makes  of  microscopes.  The  optical  parts  themselves 
are  described  below,  and  consist  of  eyepiece,  objective,  polarizer,  and  analyzer 
and,  in  some  instruments  also,  Bertrand  lens,  and  quartz  wedge  and  mica  plate. 

Besides  the  optical  parts,  there  are  also  attached  to  the  tube  an  objective 
clutch  or  a  revolving  nose-piece,  and  a  centering  device  for  the  objective. 
The  thickness  of  objective  clutch  or  nose-piece,  when  used,  must  be  added 
to  the  figures  engraved  on  the  side  of  the  instrument  to  give  the  true  mechan- 
ical tube  length. 

112.  Objective  Holders. — Objectives  may  be  attached  to  the  tube  in 
one  of  four  ways.  In  old  microscopes  it  is  necessary  to  screw  each 
objective  into  place  as  it  is  needed.  For' such  instruments  the  objective 

should  be  held  between  the  thumb 
and  first  finger  of  the  left  hand  and 
supported  at  the  side,  in  line  with  the 
tube,  by  the  bent  second  finger.  It 
should  then  be  placed  against  the 
threads  of  the  tube,  and  gently  turned 
backward  with  the  fingers  of  the  right 
hand,  until  the  beginning  of  the 
thread  drops  into  position,  after  which 
it  should  be  turned  gently  into  place. 
This  method  of  attaching  objectives 

FIG.  239.-Objective  clutch.     Natural  takes   much   #  and    there    -  fc 

size.     (Fuess.) 

danger  of   cross-threading  the  screw, 

so  that  it  is  now  almost  universally  abandoned  in  favor  of  some  rapid- 
changing  device.  These  are  of  three  types,  objective  clutch,  sliding  objec- 
tive changer,  and  revolving  nose-piece. 

Most  instrument  makers  manufacture  objective  clutches  of  a  form  similar 
to  that  shown  in  Fig.  239.*  This  is  one  of  the  most  satisfactory  appliances 
for  changing  objectives  rapidly.  It  consists  of  a  tongs  C,  made  of  steel,  which 
is  screwed  to  the  tube  at  G.  The  under  side  of  the  upper  jaw  is  beveled 
04),  and  accurately  fits  the  collar  V  which  is  screwed  to  the  objective.  To 
insert  or  remove  an  objective,  it  is  only  necessary  to  press  the  end  of  the  tongs 
with  the  left  hand  and  insert  the  objective  between  the  jaws  C  and  Z  with  the 

1  C.  Leiss:  Die  optischen  Instrumente,  Leipzig,  1899,  J87. 

See  also  Nachet's  objective-carrier,  Jour.  Roy.  Microsc.  Soc.,  1881,  661-662. 


ART.  112] 


THE  MICROSCOPE 


147 


right.  The  objective  usually,  automatically,  slides  into  proper  position  of 
perfect  centering,  where  it  is  held  by  the  lower  jaw.  It  is  well  to  give  the 
objective  a  half  turn  with  the  right  hand,  however,  to  be  certain  that  it  has 
dropped  into  its  proper  position. 

A  similar  clutch  (Fig.  308),  recently  put  on  the  market  by  Leitz  but 
not  yet  advertised,  is  so  arranged  that  the  clip  presses  against  a  slanting  bar 
on  the  objective  collar,  whereby  an  objective,  once  centered,  will  always 
return  to  the  proper  position.  A  centering  device  is  attached  to  each 
objective  collar,  the  adjustment  being  made  by  means  of  a  watch  key.  When 
once  adjusted,  the  centering  remains  perfect  for  all  objectives. 


FIG.  242. — Double  revolving  nose- 
piece.     (Bausch  and  Lomb.) 


FIG.  240.  FIG.  241. 

FIGS.  240  AND   241. — Sliding  objective  changer. 
(Zeiss.) 


FIG. 


243. — Triple  revolving  nose-piece. 
(Bausch  and  Lomb.) 


The  sliding  objective  changer,1  shown  in  Figs.  240-241,  consists  of  two 
slides,  one  of  which  is  screwed  to  the  end  of  the  tube,  the  other  to  the 
objective.  The  latter  can  be  adjusted  by  means  of  two  set-screws  so  that  the 
objective  is  accurately  centered.  This  holder,  like  the  objective  clutch  last 
described,  possesses  the  advantage  that  the  objective  is  always  inserted  with 
the  same  part  to  the  front,  and  the  centering  is  perfect. 

A  revolving  nose-piece  (Figs.  242-243)  has  a  distinct  advantage  in 
supplying  the  most  rapid  method  of  changing  objectives.  It  not  only  saves 
time,  but  is  a  safeguard  against  dropping  objectives  with  injury  to  themselves 
or  to  the  thin  section.  As  now  generally  made,  a  nose-piece  automatically 
centers  and  practically  focusses  all  objectives,  greater  or  less  length  of  collar 
making  up  for  differences  in  focal  distances.  Revolving  nose-pieces  are  made 
double  (Fig.  242),  triple  (Fig.  243),  and  quadruple,  thus  fitting  a  microscope 
with  all  the  objectives  ordinarily  needed.  The  objection  to  them  on  petro- 
graphic  microscopes  is  that  usually  all  of  the  objectives  are  not  absolutely 

1  Anon.  Zeiss1  s  objective-changer  with  slide  and  centering  adjustment.  Jour.  Roy. 
Microsc.  Soc.,  1887.  646-647. 

Anon.     Directions  for  using  the  sliding  objectiie  changer.     Zeiss'  circular,  pp.  4. 


148  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  113 

centered.  This  may  be  due  to  want  of  rigidity  in  the  attachment  to  the  tube, 
or  to  wear  in  the  device  itself.  On  a  microscope  with  simultaneously  rotating 
nicols,  they  are  a  great  convenience,  the  small  amount  which  an  objective 
may  be  out  of  center  being  of  no  importance. 

113.  Slot  for  Accessories.^-Connecting   the   tube  with  the  objective 
holder  is  a  short  collar  containing  a  slot  for  the  insertion  of  a  quartz  wedge, 
or  a  gypsum  or  mica  plate,  a  convenience  due  to  the  suggestion  of  Klein. l 
Ordinarily  a  great  deal  of  time  is  lost  in  picking  up  these  accessories  and,  in 
using  the  quartz  wedge,  in  rinding  the  proper  end  to  insert  first.     To  overcome 
this,    Johannsen2    suggested  a  permanently  attached  combination  wedge 
which  slides  in  and  out  in  a  manner  similar  to  the  Bertrand  lens.     It  is  de- 
scribed below.3    Where,  no  such  permanently  attached  accessory  is  used, 
a  sliding  plate,  in  most  microscopes,  covers,  or  is  supposed  to  cover,  the  slot 
when  not  in  use. 

114.  Centering  Device  for  Objective. — The  center  of  rotation  of  the  stage 
should  lie  on  the  axis  of  the  tube,  that  is,  it  should  coincide  with  the  intersec- 
tion of  the  cross-hairs  of  the  ocular.     For  example, 
suppose  an  object  to  be  placed  upon  the  stage  of 
the  microscope  and  the  latter  turned.     If  the  cen- 
ter of  rotation  appears  at  o'  (Fig.  244)  instead  of  at 
o,  every  other  point  in  the  slide  will  rotate  about  o' 
as  a  center.     To  make  the  center  of  rotation  coin- 
cide with  the  axis  of  the  microscope,  use  is  made 
of  two  centering  screws  which,  in  different  micro- 
scopes, are  parallel  or  diagonal  to  the  cross-hairs, 
M  ^own,  respectively,  in  Figs.  3ia  and  231,  OC. 
The  former  method  is  most  satisfactory  since  the 

adjustment  is  easier  to  make.  To  make  the  correction  shown  in  Fig.  244, 
the  center  o'  must  be  moved,  by  means  of  the  two  screws  a  and  b,  succes- 
sively through  the  distances  o'x  and  xo. 

In  some  microscopes  the  stage  is  centered  instead  of  the  objective.  This 
is  objectionable  since  it  displaces  the  entire  axis  of  the  instrument. 

If  the  centering  device  is  attached  to  the  objective  collar,  as  in  Figs.  240- 
241  and  308,  each  obiective  should  be  centered. 

1  C.  Klein:  Ueber  das  Arbeiten  mil  dem  in  ein  Polarisationsinstrument  umgewandelten 
Polari^ationsmikroskop  und  iiber  eine  dabei  in  Betracht  kommende,  vereinfachte  Methode  zur 
Bestimmung  des  Charakters  der  Doppelbrechung.     Sitzb.  Akad.  Wiss.,  Berlin,  1893  (I),  241. 

2  Albert  Johannsen :  Some  simple  improvements  for  a  petrographial  microscope.     Amer. 
Jour.  Sci.,  XXIX  (1910),  436.     The  wedge  may  be  obtained  from  Fuess  or  Leitz.     In 
ordering  it  should  be  stated  whether  the  slot  for  the  accessories  lies  parallel  or  at  45°  to 
the  principal  sections  of  the  nicols. 

3  Art.  298,  infra. 


ART.  115] 


THE  MICROSCOPE 


149 


115.  Coarse  and  Fine  Adjustment. — A  microscope  is  focussed  by  raising 
or  lowering  the  tube  with  its  optical  parts  attached.  In  general  there  are  two 
movements,  one  coarse  (gE,  Fig.  231)  and  one  fine  (fE).  In  the  instruments 
of  most  makers,  the  coarse  adjustment1  is  effected  by  a  rack  and  pinion 
(shown  to  the  right  in  Fig.  249),  while  the  fine  adjustment  screws2  are 
variously  made.  The  latter  adjustment  is  the  most  important,  for  it  is  often 
necessary  to  determine  accurately  the  amount 
of  motion;  for  example,  in  determining  the 
thickness  of  a  section. 

One  of  the  simplest  forms  of  fine  adjust- 
ment is  shown  in  Fig.  245.  The  fixed  up- 
right contains  within  it  a  spiral  spring, 
sufficiently  strong  to  carry  upward  against 
the  head,  the  weight  of  the  tube  and  its  at- 
tachments. The  upper  part  of  this  spring 
rests  against  a  cap  which,  in  its  turn,  is  kept 
down  by  a  steel  pin  attached  to  the  micro- 
meter screw  head.  The  object  of  the  pin  is 
to  transmit  the  downward  pressure  without 
setting  up  a  tendency  to  rotary  motion. 
The  pitch  of  the  micrometer  screw  is  1/2 
mm.,  and  it  has  a  total  movement  of  about 
5  mm.  The  head,  being  divided  into  50 
parts,  will  consequently  read  to  o.oi  mm. 

A  more  accurate  fine  adjustment  screw3  is 

.  .  FIG.  245- — Simple  fine  adjustment  screw. 

shown  in  r  ig.  246.     A  worm  spindle  moves  2/3  natural  size.    (Leitz.) 

a  heart-shaped  cam,  upon  which  rests  a  steel 

roller  carrying  the  weight  of  the  tube  and  its  attachments.  Since  the  periph- 
ery of  the  cam  is  formed  by  symmetrical  Archimedian  spirals,  equal  angular 
displacements  produce  equal  linear  displacements  of  the  crest  of  the  cam  (Fig. 
247),  consequently  equal  elevations  or  depressions  of  the  tube.  The  amount  of 
vertical  movement  is  indicated  by  the  micrometer  drum  (Fig.  246)  which  is 
divided  into  100  parts,  each  of  which  indicates  a  movement  of  o.ooi  mm.  or 

1  Edward    M.  Nelson:  The   track-work  coarse  adjustment.     Jour.  Roy.  Microsc.  Soc., 
1899,  256-262.     (Gives  a  history  of  the  development  of  the  coarse  adjustment.) 

2  Idem:  On  the  evolution  of  the  fine  adjustment.     Ibidem,    1899,    366-375.     "M"  (G. 
Marpmann):  Die  feine  Einstellung  der  Mikroskope.     Zeitschr.   f.   angew.   Mikrosk.,  IV 
(1898-9),  86-90. 

3  Gabriel    Lincio:    Das    neue   Leitz' sche   miner alogisc he   Mikroskopmodell    A.     Neues. 
Jahrb.,  B.B.,  XXIII  (1907),  163-186. 

E.  Leitz:  Ein  neues  Mikroskop-stativ  und  seine  feine  Einstellung.  Zeitschr.  f.  Instrum., 
XXIII  (1903),  79-81. 

Carl  Metz:  Neuere  Veroollkommnungen  der  Leitz' schen  Mikroskop-Stative.  Zeitschr.  f. 
wiss.  Mikrosk.,  XXIII  (1906),  430-439. 


150 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  116 


i  <«,  about  1/2500  in.     The  total  displacement  of  the  cam  is  3  mm.,  and  con 
tinuous  turning  of  the  screw  (JE,  Fig.  229)  produces  an  up  and  down  move- 
ment of  that  amount.     Before  measuring  thicknesses  by  means  of  this  mi- 
crometer screw,  the  index  mark  at  the  side  of  the  tube  should  be  set  half- 
way between  the  marks  55  (Fig.  231)  which  show  the  limits  of  its  motion. 


FIG.   246. — Fine  adjustment  screw.      (Leitz.) 

This  fine  adjustment  has  the  distinct  advantage  of  being  extremely 
accurate  and  of  having  no  lost  motion.  If  the  objective  should  happen 
to  be  screwed  down  upon  the  cover-glass,  it  cannot  be  forced  through,  but 

will  rest  upon  it  simply  by  the  weight  of  the 
tube,  which  is  not  great  enough  to  break  it. 

Somewhat  similar,  but  having  a  cam-like  in- 
clined plane  (£)  to  transmit  the  motion,  is  the 
fine  adjustment  shown  in  Fig.  248. 

A  fourth  type  (Fig.  249)  depends  upon  a 
lever  for  elevating  the  tube,  the  pressure  being 
produced  by  means  of  a  micrometer  screw.  In 
this  fine  adjustment,  as  in  the  last,  no  pressure 
beyond  the  weight  of  the  tube  and  its  accessories 
can  be  exerted  upon  the  cover-glass. 

116.  Sub-stage. — The  sub-stage  is  the  ring 
beneath  the  revolving  stage.  It  carries  the 

polarizer  and,  in  some  microscopes,  the  condensing  system.     It  is  described 
below1  in  connection  with  the  condensing  system. 

1  Art.  118,  infra. 


FIG.  247. — Spiral  upon  which 
the  cam  of  the  preceding  fine  adjust- 
ment screw  is  cut. 


ART.  117] 


THE  MICROSCOPE 


151 


117.  Diaphragms. — A  petrographic  microscope  should  contain  two  dia- 
phragms (besides  those  in  the  ocular  and  objective).  One,  which  is  found 
in  most  microscopes  and  called  the  lower  diaphragm,  is  placed  above  or  below 
the  analyzer,  and  is  used  to  regulate  the  amount  of  light  admitted  to  the  eye. 
The  lever  controlling  it  is  shown  in  Fig.  231,  JZ.  For  some  observations, 
especially  in  examining  colorless  objects,  too  much  light  hides  the  structure. 
Adjacent  colorless  minerals,  also,  may  only  be  distinguishable  by  means  of 
their  slightly  different  refractive  indices. 


FIG.  248. — Fine  ad j ustmentscrew. 
(Reichert.) 


FIG.  249. — Lever  fine  adjustment. 
Lomb.) 


(Bausch  and 


The  other,  or  upper  diaphragm,  is  placed  above  or  below  the  Bertrand 
lens  and  is  used  to  cut  out,  from  the  field  of  view,  interfering  minerals  when 
examining  the  interference  figure  of  a  small  grain.  The  lever  controlling  it 
is  shown  at  /,  Fig.  31  za.  This  diaphragm  is  not  found  in  old  microscopes 
nor  in  many  modern  ones,  but  is  extremely  useful  and  should  always  be  pro- 
vided. It  need  not  be  elaborate;  a  simple  perforated  slide  being  sufficient 
if  one  of  the  openings  is  extremely  small. 

Two  types  of  diaphragms  are  used  in  microscopes:  (a)  with  openings  of 
fixed  size,  (b)  with  a  changeable  opening.  There  are  various  forms  of 
the  first  type.  Among  the  older  kinds  used  were  caps,  called  cap-  or 


152 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.   117 


cylinder-diaphragms1  (Fig.  250),  having  various  sized  openings.  The  ob- 
jection to  one  of  this  style  is  that  it  takes  considerable  time  to  insert  it. 
In  some  forms  of  microscope,  it  is  even  necessary  to  remove  the  thin  section 
in  order  to  put  it  in  place. 

Another  form  is  a  simple  slide2  perforated 
with  holes  of  different  sizes.  A  special  variety,3 
with  revolving  stops,  is  shown  in  Fig.  251.  This 
diaphragm  is  attached  to  the  lower  part  of  the 
polarizer  by  means  of  three  clips  on  the  disk  LSy 
and  is  held  by  the  screw  5.  The  first  opening 
(i)  is  of  the  size  of  that  of  the  polarizer.  The 
second  (2)  is  centered  when  the  pin  of  the  catch 
(St)  drops  into  the  indentation  I,  while  all  of  those  on  the  rotating  disk  are 
centered  when  the  pin  engages  II. 

For  oblique  illumination,  such  as  is  used  in  the  Schroeder  van  der  Kolk 


FIG.  250. — Cap-   or  cylinder-dia- 
phragm.    (Zeiss.) 


FIG.   251. — Sliding  diaphragm  with  revolving  stops.     (Leitz.) 

method  of  determining  relative  refractive  indices,  it  is  only  necessary  to 
displace  the  slide  a  little  to  one  side  of  the  central  opening. 

The  first  sliding  diaphragm  was  made  by  Griffith4  in  1886.     It  is  shown 

1  Anon:  German  "cylinder-diaphragms"  and  condensers.  Jour.  Roy.  Microsc.  Soc., 
2d  series,  III  (1883),  426-427. 

2J.  Anthony:  Sliding  stage  diaphragms.  Ibidem,  1881,  520  (Describes  cardboard  or 
vellum  slips,  perforated  with  holes  of  various  sizes,  to  be  placed  on  the  stage  and  Under 
the  glass.) 

3  Anon:    Bousfield's  rotating  diaphragm-plate.     Ibidem,  1881,  523-524.     (Describes  a 
rotating  disk  without  slider,  to  be  placed  under  the  object.) 

4  E.   H.    Griffith:    Some  new  and  improied  apparatus.     Proc.   Amer.   Microsc.   Soc., 
8th  Ann.  meeting,  Cleveland,  VII  (1885),  112-114.     Review  in  Jour.  Roy.  Microsc.  Soc., 
VI  (1886),  130. 

Griffith  invented  another  form  of  lower  diaphragm,  consisting  of  circular  disks  of  metal 
with  various  shaped  holes,  or  glass  disks  covered  with  asphaltum  except  openings  of  various 
shapes.  Idem:  On  several  new  microscopical  accessories.  Proc.  Amer.  Microsc.  Soc.,  pth 
meeting,  VIII  (1886),  150-152. 


ART.  117] 


THE  MICROSCOPE 


153 


in  Fig.  252.  As  may  be  seen  from  the  illustration,  it  contains  holes  of  various 
sizes  and  shapes  which  may  be  used  centered,  or,  for  oblique  illumination, 
out  of  center.  It  is  attached  by  a  rotating  collar  below  the  nicol  so  that  the 
slot-shaped  openings  may  be  turned  in  any  direction.  A  similar  sliding  dia- 
phragm with  circular  openings  only  (Fig.  253)  was  described  by  Wright1  in  1901. 


FIG.  252. — Griffith's  sliding  diaphragm. 


FIG.  253. — Wright's  sliding  diaphragm. 


Sommerfeldt2  made  use  of  a  sliding  diaphragm  with  lath-shaped  openings 
and  placed  above  the  Bertrand  lens,  to  observe  the  interference  figures  of 
very  small  lath-shaped  crystals.  With  circular  stops,  interference  figures  of 
such  crystals  are  confused  by  those  of  surrounding 
minerals,  but  with  small  lath-shaped  slits  of  various 
sizes,  such  an  opening  may  be  selected  that  all  of  the 
surrounding  minerals  are  cut  out.  Such  a  diaphragm 
must,  however,  be  used  with  a  microscope  having 
simultaneously  rotating  nicols. 


FIG.  254. — Iris  diaphragm.  FIG.  255- — Polarizer    cas- 

ing with  nicol,  iris  diaphragm, 
and  lower  lens  of  condenser. 
3/4  natural  size.  (Fuess.) 

The  second  type  of  diaphragms  is  the  iris 3  (Fig.  254),  which  consists  of  a 
number  of  overlapping  leaves  attached  by  pins  to  the  rim  of  a  casing,  and 
having  a  central  opening  which  is  enlarged  or  diminished  by  moving  a  lever. 
This  form  is  the  most  convenient  to  use,  although,  owing  to  its  construction, 
the  opening  cannot  be  made  so  small  as  those  in  the  other  types. 

1  Fred.    Eugene    Wright:  Die  foyaitische-theralitischen,    Eruptivgestein  der  Insel  Cabo 
Frio,  Rio  de  Janeiro,  Brasilien.     T.  M.  P.  M.,  XX  (1901),  239,  footnote. 

2  E.    Sommerfeldt:  Die    mikroskopische    Achsenwinkel    bestimmung   bei   sehr    kleinen 
Kristallprdparaten.     Zeitschr.  f.  wiss.  Mikrosk.,  XXII  (1905),  361. 

3  Anon:  The  iris  diaphragm  an  old  invention.     Amer.  Jour.   Microsc.,  V  (1880),  136. 
A  description  is  quoted  from  Nicholson's  Journal  for  1804,  giving  the  construction  of  an 
iris  diaphragm.     Neither  author  nor  page  reference  is  given.     In  a  hurried  search  through 
Nicholson's  Journal,  the  original  description  could  not  be  found. 


CHAPTER  IX 


THE  MICROSCOPE     (Continued) 

The  Optical  Parts  of  a  Petrographic  Microscope 

118.  Illuminating  Apparatus.1 — The  illuminating  apparatus  of  a  micro- 
scope is  attached  to  the  sub-stage,  and  consists  of  a  mirror,  a  diaphragm,  and 
a  system  of  condensing  lenses.  In  some  of  the  older  forms  of  microscopes  the 
thin  section  had  to  be  removed,  when  changing  from  parallel  to  convergent 
light,  in  order  to  insert  the  condensing 
lens,  thus  making  necessary  a  reloca- 
tion of  the  mineral  under  examination. 
In  most  modern  microscopes  the 
change  can  be  performed  from  beneath 


FIG.  256. — Swing-out  condensing  system  and 
polarizer.     (Nachet.) 


FIG.  257. — Condensing  system  and  swing-out 
polarizer.     (Bausch  and  Lomb.) 


the  stage,  usually  by  means  of  a  lever  which  throws  in  all,  or  part,  of  the 
condensing  system;  in  some  instruments  with,  and  in  some  without,  the 
polarizer.  Each  maker,  almost,  attaches  the  condenser  to  the  microscope  in 
a  different  manner.  The  object  to  be  attained  is  the  possibility  of  rapidly  in- 
serting or  removing  it  with  as  little  disturbance  as  possible  to  other  attach- 
ments and  to  the  thin  section.2 

1  The  theory  of  illumination  is  fully  given  by  Sir  A.  E.  Wright:  Principles  of  Microscopy. 
New  York,  1907,  168-190. 

See  also  H.  E.  Fripp:  On  the  theory  of  illuminating  apparatus  employed  with  the  micro- 
scope.   Jour.  Roy.  Microsc.  Soc.,  II  (1879),  5°3~529- 

Idem:  On  daylight  illumination  with  the  plane  mirror.     Ibidem.  Ill  (1880),  742-749. 

2  See  E.  A.  Wiilfing:  Ueber  eine  Vorrichtung  zum  raschen  Wechsel  der  Beleuchtung  am 
Mikroskop.    Neues  Jahrb.,  1889  (II),  199-202. 

154 


ART.  118] 


THE  MICROSCOPE 


155 


The  upper  condenser  of  the  Fuess  microscopes  is  shown  at  6,  Fig.  233,  and 
can  also  be  seen  through  the  opening  in  the  center  of  the  stage  in  Fig.  311. 
This  condenser  can  be  removed  or  inserted  by  means  of  a  lever  (bf,  Fig.  233)  at 
the  side  of  the  rotating  stage.  Since  this  lever  does  not  always  remain  in  the 
same  position,  rotating  as  it  does  with  the  stage,  it  is  not  as  convenient  as 
though  it  were  beneath  it.  The  lower  condensing  lens  is  attached  to  the  polar- 
izer casing  (Fig.  255)  and  remains  in  position  even  when  low  power  objectives 
are  used.  Beneath  this  lens,  but  above  the  nicol  prism,  is  the  iris  diaphragm 


Frc.  258. 


PJ 


FIG.    260. 
FIGS.  258  TO  260. — Condensing  system,  iris  diaphragm,  and  polarizer.      (Leitz.) 

(i),  which  is  regulated  by  the  lever  Z.  In  the  microscope  shown  in  Fig.  311 
there  is  no  appliance  for  swinging  this  nicol  and  condenser  aside;  it  simply 
fits  snugly  in  a  collar,  from  which  it  must  be  withdrawn  by  hand. 

Certain  condensing  systems  may  be  swung  entirely  aside,  as  shown  in 
Fig.  256.  The  screw  beneath  the  pin  upon  which  the  whole  attachment 
turns  serves  to  elevate  or  depress  the  entire  system.  Above  the  nicol  is  the 
lower,  weak  condensing  lens,  which  is  used  even  in  observations  by  parallel 
light.  It  is  transformed  into  a  powerful  condenser  by  swinging  above  it  a 
second  lens  by  means  of  the  milled  head  shown.  Below  the  nicol  is  an  iris 
diaphragm.  The  nicol  itself  may  be  rotated  within  the  casing. 

Similar  in  construction  are  the  condensing  systems  shown  in  Figs.  257  and 

307- 

The  condensing  system  in  Leitz's  large  microscope1  (Figs.  230  and  231), 

1  Gabriel  Lincio:  Das  neue  Leitz' sche  miner alogische  Mikroskopmodell  A.  Neues  Jahrb., 
B.  El,  XXIII  (1907),  163-186. 


156 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  118 


shown  in  detail  in  Figs.  258-260,  may  be  raised  or  lowered  by  means  of  the 
milled  head  below  the  stage  at  BT,  Fig.  231.  Above  the  polarizer  (LF,  Fig. 
259)  is  the  iris  diaphragm  (JZ,  Fig.  258)  and  the  lower  condenser  (UC), 
both  of  which  are  carried  by  the  sliding  arm  S,  and  may  be  swung  aside.  The 
upper  condenser  (OC,  Fig.  259)  is  mounted  on  a  fork  (CB)  and  may  be  tilted 
aside  by  rotating  the  button  CH. 

The  mirror1  is  used  to  reflect  the  light  from  the  source  to  the  object,  and 
is  attached  to  the  sub-stage  by  means  of  the  mirror  bar.  It  may  be  inclined 
in  any  direction  by  means  of  its  various  pivots,  and  may  be  rotated  so  that 
either  side  is  uppermost.  A  plane  mirror  forms  one  side  and  a  concave  the 
other,  the  former  being  used  with  low  magnifications,  where  a  comparatively 
weak  light  is  sufficient,  while  the  latter  is  used  for  higher  magnifications, 
concentrating  the  light  by  converging  the  rays  included  within  an  angular 
aperture  of  about  40°.  For  very  high  magnifications  and  for  phenomena  to 
be  observed  in  convergent  light,  the  condensing  lens,  already  described,  is 
used. 


FIG.  261. — Parallel  rays  of  light, 
plane  mirror,  no  condenser. 


FIG.  262. — Parallel  rays,  concave 
mirror,  no  condenser. 


When  parallel  rays,  such  as  those  reflected  from  the  sky,  are  used,  they  are 
reflected  from  the  plane  mirror  with  a  slight  loss  of  intensity  (Fig.  261). 
From  the  concave  mirror  they  are  reflected  with  increased  intensity,  the  rays 
coming  together  at  the  focal  point  (Fig.  262).  For  any  source  of  light  nearer 
the  instrument,  the  focal  distance  is  greater  (Fig.  263)  until  it  may  be  twice  as 
great  when  the  source  is  quite  near.  To  adjust  this  distance,  the  mirror  is 
attached  to  the  mirror-bar  by  means  of  a  sliding  sleeve  (Figs.  307  to  324). 

The  condensing  lens,  being  constructed  so  that  its  focus  is  some  distance 
above  its  upper  surface,  should  always  have  the  plane  mirror  (Fig.  264)  used 
in  connection  with  it.  The  concave  mirror  causes  the  rays  to  converge  too 
low  down  (Fig.  265)  to  give  the  best  results,  which  are,  clearly,  to  be  obtained 
when  the  condenser  is  in  focus.  This  may  be  accomplished  by  placing  upon 

1  See  Edward  M.  Nelson:  Construction  of  silvered  lens  mirrors.  Jour.  Roy.  Microsc. 
Soc.,  1894,  254-260. 


ART.  118] 


THE  MICROSCOPE 


157 


the  stage  a  thin  section  and  focussing  upon  it  with  a  low  power  objective. 
The  plane  mirror  is  then  turned  until  some  moderately  distant  object, 
such  as  a  window  bar,  tree,  etc.,  appears  in  the  field,  after  which  the  condenser 
is  racked  up  or  down  until  the  image  of  this  object  becomes  perfectly  sharp. 
After  the  condenser  is  in  focus,  the  image  by  which  it  was  focussed  should  be 
removed  from  the  field  of  the  microscope  by  a  slight  rotation  of  the  mirror. 


FIG.  263. — Divergent  rays,  concave  mirror,  no  condenser. 

Since  focussing  the  condenser  is  the  process  of  adjusting  its  elevation  in 
relation  to  the  surface  of  the  slide,  it  is  not  necessary,  ordinarily,  to  refocus 
it  for  different  power  objectives,  but  if  the  glasses  upon  which  the  various 
sections  are  mounted  are  greatly  different  in  thickness,  it  may  be  necessary 


FIG.  264. — Correct  method  of  illu- 
mination. Parallel  rays,  plane  mirror, 
and  condenser. 


FIG.  265. — Incorrect  method  of 
illumination.  Parallel  light,  concave 
mirror,  and  condenser. 


do  so.  In  some  microscopes  there  is  not  much  leeway  in  regard  to 
focussing,  the  image  being  sharp  only  when  the  condenser  rests  against  the 
bottom  of  the  mount. 


158 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  119 


POLARIZING  PRISMS 

119.  Introduction. — Many  of  the  most  important  characteristics  of  crys- 
tals are  determined  by  polarized  light,  and  the  appliance  for  producing  it 
forms   an   important  adjunct  to  the  petrographic  microscope.     We  have 
already  seen1  what  polarized  light  is,  and  have  learned  that  it  may  be  pro- 
duced by  reflection,  single  refraction,  absorption,  or  double  refraction.     In  a 
few  simple  microscopes  (Figs.  723  and  724)  light  is  polarized  by  reflection, 
but  in  the  usual  petrographic  microscopes,  some  form  of  calcite  (Iceland  spar) 
prism  is  used. 

120.  Nicol  Prism  (1828). — The  first  polarizing  prism  of  Iceland  spar  was 

invented  by  W.  Nicol2  in  1828,  and,  after  him,   such 
prisms  are  called  nicol  prisms  or  simply  nicols. 

In  constructing  this  prism,  a  cleavage  rhombohedron 
of  calcite,  about  three  times  as  long  as  it  is  broad,  is 
used.  It  is  cut  diagonally  across  in  a  plane  parallel  to 
the  long  diagonal  of  the  end  faces  (BD,  Fig.  266), 
along  the  line  represented  by  A'G'  in  Fig.  267,  making 
an  angle  of  22°  with  the  edge  AE.  The  natural  faces 
ABCD  and  EFGH,  -Fig.  266,  make  angles  (CAE  and 
ACG,  Fig.  267)  of  109°  7'  and  70°  53'  with  the  edges 
AE  and  CG,  and  these  faces  are  ground  down,  at  A  and 
G,  until  they  form  angles  of  112°  and  68°  (CA'E  and 
A'CG,  Fig.  267)  with  AE  and  CG,  making  the  angle 
GAfC  =  go°.  The  two  pieces,  after  being  polished  upon 
the  cut  faces,  are  cemented  in  their  original  position 
by  Canada  balsam,  blackened  on  the  faces  which  are 
vertical  in  Fig.  266,  and  set  in  a  cork  and  metal  casing. 
If,  now,  a  ray  of  light  falls  upon  one  end  of  the  nicol 
prism,  it  will  be  broken  up,  by  the  double  refraction,  into 
two  rays,  one  of  which  will  be  more  refracted  than  the  other.  If  the  light 
falls  upon  the  lower  surface  (Fig.  268)  at  such  an  angle  that  the  extra- 
ordinary ray  travels  in  a  direction  parallel  to  the  long  edges,  it  will 

1  Art.  42,  supra. 

2  William  Nicol :  On  a  Method  of  so  far  increasing  the  Divergence  of  the  two  Rays  in 
Calcareous-spar  that  only  one  image  may  be  seen  at  a  time.     Edinb.  New  Phil.  Jour.,  VI 
(1828-9),  83-94. 

Idem:  Notice  concerning  an  Improiement  in  the  Construction  of  the  Single  Vision  Prism 
of  Calcareous  Spar.  Ibidem,  XXVII  (1839),  332-333. 

E.  Sang:  Investigation  of  the  action  of  Nicol's  polarizing  eye-piece.  Read  Feb.  20,  1837. 
Published  in  Proc.  Roy.  Soc.  Edinburgh,  XXXIII  (1890-1),  323-336. 

Prof.  Tait:  Note  on  Dr.  Sang's  paper.     Ibidem,  337-340. 

M.  Spassky:  Note  iiber  das  Nicol'sche  Prisma.     Pogg.  Ann.,  XLIV  (1838),  168-176. 

K.  Fuessner:  Ueber  die  Prismen  zur  Polarization  des  Lichtes.  Zeitschr.  f.  Instrum., 
IV  (1884),  41-50. 


FIG.  266. — A  cleavage 
rhombohedron  of  Iceland 
spar. 


ART.  120] 


THE  MICROSCOPE 


159 


make  an  angle  of  63°  44'  with  the  c  axis  of  the  crystal  until  it  strikes 
the  film  of  balsam.  The  direction  of  vibration  of  the  extraordinary 
ray  in  a  crystal  of  calcite,  as  we  have  already  seen,1  is  in  the  plane  containing 
crystallographic  c  and  the  ray,  consequently  in  the  plane  ACGE,  Fig.  266, 


FIG.  267. — The  form  of  a  nicol  prism. 


FIG.  268. — Passage  of  light  through 
a  nicol  prism. 


The 


and  may  be  represented  by  the  short  lines  along  the  ray  in  Fig.  268. 
index  of  refraction  of  this  ray,  by  calculation,  is  found  to  be  1.5159. 

From  equation  (9),  Art.  53,  we  can  obtain  the  index  of  refraction  of  the  ray  in 
question. 

i      sinV  ,  cosV 


where  ei  is  the  index  of  refraction  of  the  wave  whose  normal  makes  an  angle  of  ^ 
with  the  c  axis.     Substituting,  in  this  equation,  the  values  of  the  principal  indices 

1  Art.  48,  supra. 


160  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  120 

of  refraction  of  calcite,  €=1.4864,  w=  1.6585,-  and  the  value  of  ^=63°  44',  and 
solving  for  ci  we  have 

Cl  =  1.5159,  (i) 

the  value  of  the  index  of  refraction  of  the  ray. 

It  was  assumed  above  that  the  ray  of  light,  falling  upon  the  lower  face  of 
the  nicol,  made  such  an  angle  that  the  extraordinary  ray  passed  through  it 
parallel  to  its  long  sides.  In  order  to  do  this,  as  will  be  shown,  the  light, 
coming  from  below,  must  make  an  angle  of  34°  36'  with  the  normal  to  the 
lower  face.  At  this  angle  of  incidence,  the  ordinary  ray  will  make  an  angle 
of  20°  i'  with  the  normal,  or  69°  59'  with  the  balsam  film. 

From  equation  (8)  Art.  37,  we  know  that 

sin  i  . 


where  i  is  the  angle  of  incidence  and  rf,  the  angle  of  refraction  of  the  extraordinary 
ray.  Now  since  the  extraordinary  ray  assumed  above  passes  straight  through  the 
crystal,  it  forms  an  angle  of  22°  with  the  balsam  film,  and  since  the  base  of  the 
crystal  is  at  right  angles  to  this  film,  the  extraordinary  ray  makes  the  same  angle 
with  the  normal  to  the  base.  That  is,  22°  is  also  the  angle  of  refraction  of  the  extra- 
ordinary ray  (re  =2  2°,  Fig.  268). 

Substituting  in  equation  (2)  this  value  and  the  value  for  «i  (  =  1.5159)  found 
above,  we  find 

sin  i 


sin  *  =  .  5678,  (3) 

*  =  34°36',  (4) 

which  is  the  angle  made  by  the  incident  light  with  the  normal  to  the  lower  face  of 
the  prism. 

The  angle  which  the  ordinary  ray  makes  after  passing  into  the  crystal  is  obtained 
from  the  equation 

sin  i  ,  x 

-  --  =  o)  (5) 

sm  ru 

where  i  is  the  angle  of  incidence,  and  ra,  the  angle  of  refraction  of  the  ordinary  ray. 
The  index  of  refraction  of  the  ordinary  ray  is  the  same  in  every  direction,  there- 
fore w=  1.6585.     Substituting  this  value  and  the  value  for  sin  i  from  equation  (3) 
in  equation  (5),  we  have: 

0.5678 


ra=20°i',  (6) 

which  is  the  angle  of  refraction  of  the  ordinary  ray.  The  normal  being  parallel 
to  the  balsam  film,  this  is  also  the  angle  between  the  ordinary  ray  and  the  balsam, 
which  makes  the  angle  of  incidence  at  this  film 

900-200i/  =  69°  59'.  (7) 


ART.  120]  THE  MICROSCOPE  161 

Let  us  now  see  what  takes  place  when  the  extraordinary  and  ordinary 
rays  strike  the  balsam  film. 

The  index  of  refraction  of  old  balsam  is  approximately  1.54.  That  of 
the  extraordinary  ray  is  1.5159  (Eq.  i);  consequently,  passing  from  a  rarer 
to  a  denser  medium,  the  ray  will  be  transmitted,  and,  upon  entering  the  second 
half  of  the  prism,  will  resume  its  original  course  along  a  line  parallel  to  the 
long  sides  of  the  prism  and  pass  out  at  D. 

The  ordinary  ray,  with  an  index  of  1.6585,  strikes  a  medium  with  an  index 
of  1.54  at  an  angle  of  69°  59'.  Now  the  critical  angle  for  the  ordinary  ray, 
in  passing  from  calcite  to  balsam,  is  68°  13',  and  the  ray  in  question,  falling 
upon  the  separating  plane  at  a  greater  angle,  is  totally  reflected,  passes  aside 
at  D',  and  is  absorbed  by  the  black  paint  with  which  the  sides  are  coated. 

The  critical  angle  of  the  ordinary  ray   (index  of   refraction  =    1.6585)  upon 
striking  the  balsam  (index  of  refraction  =  1.54)  is  obtained  as  follows: 
In  calcite  we  have 

S!n  *,.„•-!.  6585, 

sin  r 
and  in  balsam 

sin  i         „ 

sinr"  =  W  =I-54' 

Dividing  the  former  by  the  latter,  we  have 

sin  i 

sin  r'  _sin  r"  _  i .  6585 
sin  i      sin  r'       i  -  54 
sin  r" 

The  critical  angle  between   the  two  will  be   reached  when  r"  =  9o°  (Art.  41) 
whereby  the  sine  of  r"  would  equal  i. 
Our  equation  therefore  becomes 

i      =  i^SJJS 
sin  r'~   1.54  ' 

sin  r'  =0.9286,  (8) 

r  =  68°  13', 
which  is  the  value  of  the  critical  angle  between  the  two  media. 

We  see,  from  this  demonstration,  that  a  ray  of  light,  falling  upon  the  base 
of  a  nicol  prism  and  making  an  angle  of  12°  36'  (34°  3 6 '-2 2°)  with  its  long 
direction,  will  be  broken  up  into  two  rays,  one  of  which,  the  extraordinary, 
will  pass  through  it  in  a  straight  line  (except  for  a  slight  displacement  by  the 
film  of  balsam),  while  the  other,  the  ordinary,  will  be  totally  reflected.  The 
extraordinary  ray,  at  the  same  time,  is  polarized,  and,  upon  emerging,  has 
its  vibrations  in  a  plane  at  right  angles  to  the  film  of  balsam. 

If  one  looks  through  a  nicol  prism  at  a  bright  surface,  such  as  a  white 
cloud,  it  will  be  seen  that  beyond  a  certain  distance  from  the  center,  and 
11 


162 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  120 


Separated  from  it  by  a  blue-black  band,  the  field  is  dark.  This  is  due  to  the 
fact  that  beyond  that  distance  the  extraordinary  ray  also  falls  on  the  balsam 
film  at  an  angle  greater  than  its  critical  angle,  consequently  it  also  is  totally 
reflected.  On  the  other  side  of  the  prism  there  is  an  area  of  great  brightness, 
separated  from  the  central  part  by  a  narrow  series  of  colored  bands.  This 
area  represents  the  region  where  the  ordinary  as  well  as  the  extraordinary 


FIG.  269. — Nicol  prism  showing 
the  limiting  rays  beyond  which 
the  light  is  not  plane  polarized. 


FIG.  270. — Nicol  prism. 


ray  passes  through.  The  internal  angle  between  the  rays  forming  these 
limits  is  about  14°,  which  is  increased  to  25°  by  refraction  as  they  pass  into 
the  air.  The  latter  angle  is  called  the  useful  opening  angle  of  the  nicol. 

To  determine  the  opening  angle  of  the  nicol,  the  problem  becomes  that  of 
finding  the  angles  at  which  the  light  must  enter  the  prism  so  that  the  ordinary 
and  the  extraordinary  rays  may  reach  the  balsam  film  at  their  respective  critical 
angles. 


ART.  120]  THE  MICROSCOPE  163 

The  critical  angle  for  the  ordinary  ray  is  68°  13',  and  the  angle  which  it  makes 
with  the  normal  to  the  base  of  the  prism  (Fig.  269)  is  equal  to 


whereby 

sin  i 


sin  21    47 
sin  2=1.6585X0.3711  =0.61547, 
*  =37°59/, 

which  is  the  angle  of  incidence  at  the  lower  surface  when  the  ordinary  ray  is  at  the 
critical  angle. 

Since  the  normal  lies  at  an  angle  of  22°  with  the  line  parallel  to  the  long  sides 
of  the  crystal,  this  ray  will  form  an  angle  of  37°  59'  —  2  2°  =15°  59'  with  it.  This 
angle  is  the  limiting  value  of  the  nicol  in  this  direction,  since  the  ordinary  as  well 
as  the  extraordinary  ray  of  light,  entering  from  beyond  it,  will  pass  through  the  bal- 
sam instead  of  being  reflected. 

-The  determination  of  the  incident  angle  for  the  limiting  value  in  the  othei  direc- 
tion is  more  complicated,  since  the  critical  angle  must  be  determinec^fpr  a  ray  whose 
index  of  refraction  changes  with  its  direction  of  transmission  (Fig.  270). 

Let  i  =  the  angle  of  incidence  of  the  limiting  extraordinary  ray. 
r  =  its  angle  of  refraction. 

90°—  r  =  the  critical  angle  against  the  balsam  film. 
v>V/  =  its  index  of  refraction. 

*?  =  41°  44;+r  =  the  angle  between  crystallographic  c  and  the  normal 
to  the  wave  front  of  the  ray. 

Determining  the  critical  angle'tas  was  done  in  equation  (8),  we  have: 

sin  (9o°-f)  =  cos  r  =^,  or  €,,  =  ^~-  d) 

sin  i  t  x 

Also  --=«:„.  (2) 

sin  r 

and  equation   (9) 

i       sin2  <p     cos2  <p 


Combining  equations  (i)  and  (3) 
(i-54)2 


cos2r       w2  sin2  ?+  e2  cos2  <p' 
Reducing 

<«V  cos2  r  =(1.54)2  o>2  sin2s?+(i.54)2e2cosV. 

Substituting  sinV=  i  —  cos2^,  and  the  values  of  t,w  ,  and  <?, 

i. 216  cos2  r  =  i. 294-. 255  cos2  (41°  44'  +  r), 


164  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  121 

from  which 

r  =  8°26'.  (4) 

Substituting  this  value  in  (i) 


Substituting  values  from  (4)  and  (5)  in  (2), 

sin  i  =1.5568  sin  8°26', 

*=i3°i2'.  (6) 

The  critical  angle,  therefore,  is 

9o0-r  =  9o0-i30i2'  =  8i°34'.  (7) 

The  inclination  of  the  incident  ray  to  the  vertical  may  be  seen,  in  Fig.  269  to  be 

220-i3°i2'  =  8048'.  (8) 

The  total  serviceable  opening  of  the  nicol  is  the  angle  between  the  entrance  angles 
of  the  rays  which  produce  the  limiting  rays,  upon  one  side  where  the  ordinary 
reaches  its  critical  angle,  and  upon  the  other  where  the  extraordinary  reaches  its 
critical  angle.  ^As  shown  above,  this  angle  is  equal  tA 


For  certain  purposes,  such  as  analyzers  of  microscopes,  the  sloping 
ends  of  the  nicol  prism  are  objectionable,  since  they  both  diminish  the  light  by 
reflection,  and  displace  the  object.  In  certain  cases  it  may  be  desirable  to 
use  a  prism  having  a  lesser  length  in  relation  to  its  breadth,  or  having  a  larger 
opening  angle  than  that  of  the  nicol.}  For  these  reasons  many  modifications 
have  been  proposed,  among  which  the  more  important  are  the  following: 

121.  Sang  Prism  (1837).  —  In  a  paper  read  before  the  Royal  Society, 
Edinburgh,  in  1837,  but  not  published  until  1891,  Sang1  discussed  the  prin- 
ciple of  the  Nicol  prism,  and  calculated  the  most  effective  angle  at  which  it 
can  be  cut.  LHe  suggested  that  a  prism  might  be  constructed  so  that  the  ordi- 
nary ray  would  pass  and  the  extraordinary  be  reflected.  To  make  this  pos- 
sible, a  plate  of  calcite  must  be  placed  between  two  wedges  of  a  medium 
having  a  higher  index  of  refraction  than  that  mineral.  Such  a  prism  is  shown 
in  Fig.  271,  in  which  AB  is  a  plate  of  Iceland  spar  cut  at  right  angles  to  the 
optic  axis,  and  ABD  and  ABC,  two  wedges  of  glass  having  a  refractive  index 
of  1.655.]  The  angle  ABD  must  be  26°  10'  19"  in  order  that  the  extraordinary 
ray  within  the  prism  may  suffer  total  reflection,  whereby  only  the  ordinary  ray 
of  all  light  entering  between  A  and  D,  will  pass  through.  Between  D  and  B^ 
both  rays  will  pass.  The  angle  BDA  is  52°  50'  35"  and  BAD,  100°  59'  06". 

1  E.   Sang:   Investigation  of  the  action  of  Nicol'  s   polarizing  eye-piece.     Read   Feb.  20, 
1837,  published  in  Proc.  Roy.  Soc.  Edinburgh,  XVIII  (1890-1),  323-336. 
Prof.  Tait:  Note  on  Dr.  Sang's  paper.     Ibidem,  337-340. 


ART.  123] 


THE  MICROSCOPE 


165 


122.  Foucault  Prism  (1857). — The  prism  constructed  by  Foucault1 
consists  of  a  cleavage  rhombohedron  of  Iceland  spar  with  natural  faces,  and 
so  cut  that  the  section  makes  angles  of  51°  and  58°  7'  with  these  faces  (Fig. 
272).  The  two  parts  are  not  cemented  but  are  separated  by  a  film  of  air. 
This  has  the  effect  of  increasing  the  difference  between  the  indices  of  refrac- 
tion between  the  calcite  and  the  film,  consequently  of  decreasing  the  critical 
angle.  The  smaller  the  critical  angle,  the  shorter  will  be  the  required  prism; 
here  it  bears  a  ratio  of  about  1.5  to  i,  length  to  width. 

The  ordinary  ray  reaches  the  film  of  air  at  an  angle  greater  than  its  critical 
angle  (C.A.  =37°  14')  and  is  totally  reflected.  The  extraordinary  ray  is 


FIG.  271. — Sang  prism. 


FIG.  272. — Foucault  prism. 


transmitted  in  part,  but  owing  to  the  reflecting  surfaces  at  the  film,  about 
10  per  cent,  is  lost.  The  critical  angle  of  the  extraordinary  ray  is  42°  23'; 
there  is,  therefore,  an  angle  of  pnly  5°  9'  between  these  limiting  values,  equal 
to  an  angle  of  about  8°  upon  the  emergence  of  the  rays  in  air. 

The  advantage  of  the  Foucault  prism  is  its  economy  in  the  use  of  Iceland 
spar.  Its  disadvantage  is  the  loss  of  transmitted  light  and  its  small  opening 
angle,  which  prevents  its  use  for  convergent  light. 

123.  Hartnack-Prazmowski  Prism    (1866). — The  prism  of   Hartnack- 

1  Leon  Foucault:  Notneau  polariseur  en  spath  d1  Island.  Comptes  Rendus,  XLV  (1857), 
238-241. 

Anon:  Neuer  Polarisator  von  Kalkspalh.  (Review  of  preceding.)  Pogg.  Ann.,  CII 
(1857),  642-3. 


166 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  123 


Prazmowski1  gives  a  wide  opening  angle,  varying  between  28°  and  42°  with 
the  kind  of  cementing  material  and  the  angle  at  which  the  ends  are  cut. 
As  originally  described  the  prism  was  rectangular  (Fig.  273)  and  was  so  cut 
that  the  optic  axis  of  the  crystal  lay  at  right  angles  to  the  plane  joining  the 
two  parts.  >.  The  angles  which  this  intersecting  plane  made  with  the  end  faces 
for  various  cementing  media  are  given  in  the  following  table. 


Angle 

Cementing 
material 

Value 
of  n 

between 
end  faces 

Field 

Length 

Interior 
angle 

and  film 

Canada  balsam  

1-549                79° 

33° 

5-2 

20°   54' 

Copaiva  balsam  .... 

I-507                76°  30' 

35° 

3-7 

24°  42' 

Linseed  oil  

1.485 

73°  30' 

35° 

3-4 

26°   24' 

Poppy  oil  i  .  463 

7i° 

28° 

3-° 

17° 

FIG.  273, 
Prazmowski 
pared  with 
nicol. 


This    prism    was    re-calculated    by    Fuessner2    who 
found  that  its  most  advantageous  form  was  produced 
when  the  end  faces  formed  an  angle  of  76°  5'  with  the 
film  of  linseed  oil.     Such  a  prism  gives  a  field  in  air  of 
41°  54'  and  the  ratio  of  its  length  to  width  is  4.02. 
\    When  the  extraordinary  ray  is  reflected  parallel  to  the 
film,  the  ordinary  forms  an  angle  of  26°  22'  with  it,  re 
being  13°  55'  and  rw  12°  27'.  If  a  balsam  film  is  used 
re  =  io°  iS'  and  rw   =11°  30'.     The  prism  may  be  con- 
prism,  com-  siderably  shortened,  at  the  expense  of  the  field  of  view, 
an  ordinary  by  decreasing  the  angle  between  the  film  and  the  end 
faces. 


Field 

Angle  between 
end  faces 
and  film 

Ratio  length 
to  width 

4i°  54'                    76°    5' 

4.04 

30° 

72°  37' 

3-5i 
3-19 

All  prisms  with  linseed  oil  films. 

20° 

69°  39' 

2.70 

[The  advantages  of  this  prism  are  its  square  ends  and  its  high  opening 
angle,  which  throws  the  blue  fringe  far  to  one  side.  Its  disadvantages  are  its 
wastefulness  of  spar,  its  great  length  compared  with  its  width,  and  the  fact 

1Hartnack  et  Prazmowski:  Prisme  polarisateur.  Ann.  Chim.  et  Phys.,  4  sen,  VII 
(1866),  181-189. 

Deleuil:  Prisme  polarisateur  de  MM.  Hartnack  et  Prazmowski.  Comptes  Rendus, 
LXII  (1866),  149-150. 

Review  of  preceding:  Polarisations prisma  von  Hartnack  und  Prazmowski.  Pogg.  Ann., 
CXXVII  (1866),  494-496. 

2  K.  Fuessner:  Ueber  die  Prismen  zur  Polarisation  des  Lichtes.  Zeitschr.  f.  Instrum., 
IV  (1884),  41-50. 


ART.  126] 


THE  MICROSCOPE 


167 


that  linseed  oil  dries  in  time,  causing  its  index  of  refraction  to  increase  and 
producing  bubbles  in  the  film. 

124.  Talbot  Prism  (1872). — In  order  to  reduce  the  amount  of  Iceland  spar 
necessary  to  make  a  nicol  prism,  Talbot,1  in  1872,  constructed  one  in  which 
one-half  was  replaced  by  a  prism  of  glass.     No  further  description  was  given 
of  it  except  that  "either  end  could  be  held  foremost,"  probably  meaning 
when  used  as  an  analyzer. 

125.  Glan  Prism  (1880). — The  Glan2  prism  is  much  shorter  than  any  of 
the  preceding,  the  ratio  of  length  to  breadth  being  theoretically  0.831,  though 
in  practice  it  is  customary  to  let  the  two  pieces  project  beyond  the  cut  sur- 
face as  shown  in  Fig.  274,  making  the  ratio  0.924  to  1.141. 


FIG.  274- — Glan  prism  in  section. 


FIG.   275. — Glan  prism  in  perspective. 


fThe  prism  differs  irom  those  described  above  in  that  its  optic  axis  lies 
in  the  plane  of  separation  and  at  right  angles  to  the  side  faces;  consequently 
parallel  to  the  end  faces.  The  separating  film,  which  is  of  air  1/2  mm.  thick, 
forms  an  angle  of  50°  17'  with  the  sides.  While  this  prism  has  the  advantage 
of  shortness,  it  has  the  disadvantage  of  having  an  opening  angle  of  only  ap- 
proximately 8°,  and  likewise  of  causing  considerable  loss  of  light  on  account 
of  the  separating  air  film,.; 

This  prism  is  sometimes  called  Glan-Foucault  since  it  embodies  some  of 
the  principles  of  the  Foucault  prism  described  above. 

126.  Thompson  Prisms  (1881  and  1886). — In  Professor  Thompson's 
1881  prism3  the  opening  angle  is  about  35°.  VThe  external  form  is  the  same  as 
that  of  the  nicol  prism,  but  cryst^llographic  c  lies  at  right  angles  to  the  axis 
of  the  prism  and  in  the  balsam  film]  (Fig.  276).  By  this  means  the  blue 
fringe  is  removed  from  the  field.  \  Thompson  suggested  cutting  the  end 
faces  more  oblique,  which  would  reflect  the  ordinary  ray  farther  and  increase 

1  H.  F.  Talbot:    On  the  nicol  prism.     Proc.  Roy.  Soc.  Edinburgh,  VII  (1872),  468-470. 
Compare  the  prisms  of  Leiss  (1897)  and  of  Lommel  (1898),  described  below. 

2  P.  Glan:  Ueber  einen  Polar  is  ator.     Carl's  Repertorium,  XVI  (1880),  570-73. 
Idem:    Nachtrag  zum  Polarisator.     Ibidem,  XVII  (1881),  195. 

3  Silvanus  P.  Thompson:    On   a  new  polarizing  prism.     Phil.  Mag.  5  ser.  XII  (1881), 
349-35L 

Idem:  Same  title.     Rept.  Brit.  Asso.  Adv.  Sci.  1881,  563-564. 


168 


MANUAL  OF  PETROGRAP1IIC  METHODS 


[ART.  127 


the  opening  angle.  It  would,  however,  decrease  the  amount  of  light  by 
reflection,  and  increase  the  distortion  of  the  field.  There  is  much  waste  in 
cutting  this  prisrru\  % 

In  1886,  Professor  Thompson1  suggested  another  kind  of  prism  which 
he  called  a  " reversed  Nicol,"  and  which  possesses  certain  advantages  and 
does  not  add  much  to  the  cost.  The  broken  line  in  Fig.  277  represents  a 


FIG.  276. — Thompson's  earlier  prism. 


PIG.   277- — Thompson's  reversed  nicol. 


nicol  prism  as  usually  cut,  the  solid  lines,  Thompson's  modification.  Each 
end  is  first  ground  down  about  40°  from  the  natural  faces,  leaving  an  angle 
of  69°  with  the  long  edges.  It  is  then  cut  across  at  an  angle  of  22°  with  the 
same  edges  and  cemented.  The  result  is  a  shortened  and  reversed  nicol 
which  possesses  the  advantage  of  having  the  crystallographic  axis  nearly  at 
right  angles  to  the  direction  of  transmission  of  the  light  and  nearly  at  right 
angles  to  the  balsam  film,  with  the  result  that  the  blue  fringe  is  thrown  far- 
ther back,  giving  a  prism  which  is  shorter,  and  with  a  field  equally  wide  or 
wider  than  the  ordinary  nicol. 

127.  Fuessner  Prisms  (1884). — Fuessner,2  in  1884,  invented  a  number  of 

1Idem:  Notes  on  some  new  polarizing  prisms.  Phil.  Mag.,  5  ser.,  XXI  (1886), 
476-480. 

2  K.  Fuessner:  Ueber  die  Prismcn  zur  Polarisation  des  Lichtes.  Zeitschr.  f.  Instrum., 
IV  (1884),  47-49. 

Review  of  preceding  in  Jour.  Roy.  Microsc.  Soc.,  IV  (1884),  456-462. 

See  also  Ph.  Sleeman:  Dr.  Fuessner'' s  new  polarizing  prism.  Nature,  XXIX  (1884), 
5I4-5I7. 


ART.  128]  THE  MICROSCOPE  169 

new  polarizing  prisms  designed  to  give  a  large  field  and  at  the  same  time  be 
less  expensive  than  the  ordinary  nicols.  His  prisms  are  similar  to  those  sug- 
gested by  Sang1  in  1837  though  not  published  until  1891. 

Fuessner  described  a  prism  of  glass  cut  diagonally  across  and  reunited 
after  the  insertion  of  a  thin  plate  of  calcite.  The  cement  used  must  have  the 
same  index  of  refraction  as  the  glass,  and  both  must  equal  the  greatest  index 
of  the  calcite.  The  directions  of  greatest  and  least  ease  of  vibration  must  lie 
in  a  plane  normal  to  the  cut  section  of  the  glass.  Since  calcite  is  uniaxial, 
any  section  may  be  so  placed,  and  cleavage  pieces  can  be  obtained  easily.  A 
calcite  prism,  4.25  times  as  long  as  it  is  wide,  when  made  on  this  principle, 
has  an  opening  angle  of  44°. 

But  other  crystals  than  calcite  may  be  used.  All  that  is  necessary  is  that 
they  be  colorless  and  transparent.  If  the  difference  in  the  indices  in  the  two 
directions  is  greater  than  that  of  calcite,  the  field  will  be  larger  and  the 
prism  shorter.  A  prism  constructed  of  glass  and  a  plate  of  sodium  nitrate, 
which  has  indices  05  =  1.587  and  6  =  1.336,  gave  a  field  of  56°  with  a  ratio  of 
length  to  breadth  of  3.34.  In  this  prism  a  cement  of  Damar  resin  in 
monobromnaphthalene  was  used.  Damar  resin  consists  of  two  resins,  one 
of  which  is  soluble  in  alcohol.  The  residue  is  very  brittle  and  colorless, 
and  has  an  index  of  refraction  of  1.549.  If  one- third  of  its  volume  of 
monobromnaphthalene  be  added,  a  viscid  cement  with  an  index  of  1.58  is 
produced.  No  satisfactory  cement,  with  an  index  as  high  as  that  of  calcite, 
was  found,  although  tolu  balsam  in  monobromnaphthalene  gives  an  index  of 
refraction  of  1.62.  This,  however,  on  account  of  its  lower  refractive  index, 
cut  the  field  down  to  34°.  If  the  prism  is  fitted  into  a  glass  tube,  a  liquid 
film  of  monobromnaphthalene,  which  has  an  index  exactly  equal  to  co  of 
calcite,  may  be  used. 

128.  Bertrand  Prisms  (1884-1885). — Bertrand2  described  a  number  of 
prisms  quite  similar  to  those  of  Fuessner.  His  flint  glass  prism,  with  a  re- 
fractive index  of  1.658,  is  cut  in  a  plane  inclined  76°  43'  8"  to  the  end  faces, 
and  recemented  after  having  had  inserted  between  the  two  pieces  a  cleavage 
plate  of  calcite.  It  differs  from  that  of  Fuessner  in  the  orientation  of  the  cal- 
cite, which  has  its  optic  axis  parallel  to  the  end  faces  (Fig.  278).  The  ordi- 
nary ray,  consequently,  will  be  the  one  which  passes  through.  The  cement 
must  have  an  index  of  refraction  of  1.658  or  greater.  The  resulting  prism 
has  a  length  equal  to  the  Hartnack-Prazmowski  but  a  field  of  44°  46'  20".  • 

I0p.  cit.,  Art.  121. 

2  Emile  Bertrand:  Sur  un  nouwau  polarisaleur.  Comptes  Rendus,  XCIX  (1884), 
538-540.  . 

Idem:    Sur  di/erents   prism?s   polarisateurs.     Bull.  Soc.  Min.    France,  VII    (1884), 

339-345- 

Idem:  Ueber  vtrschiedem  Polarisationsprismen.  Beiblatter  zu  Wiedem.  Ann.,  IX 
(1885),  428-430. 


170 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  129 


Ot>ti 


The  advantage  of  this  prism  is  its  cheapness,  since  but  a  very  small  amount 
of  calcite  is  used. 

Another  prism  is  made  of  flint  glass,  with  an  index  of  1.586,  cut  at 
74°  with  the  end  faces,  and  having  a  thin  plate  of  sodium  nitrate  inserted  with 
its  optic  axis  parallel  to  the  end  faces.  The  cement  must  have  an  index  of 
1.568  or  more.  The  field  of  view  is  53°. 

A  third  prism  (Fig.  279)  consists  of  cal- 
cite, with  the  optic  axis  nearly  parallel  to 
the  end  faces,  and  cut  on  a  plane  making 
an  angle  of  76°  to  77°  with  these  ends.  It  is 
cemented  with  Canada  balsam  after  having 
had  inserted  a  thin  glass  plate  with  an  index 
of  1.483.  This  prism  is  like  the  Hartnack- 
Prazmowski  in  its  action  without  having 
the  bad  properties  of  the  linseed  oil  film  of 
the  latter. 

Bertrand  also  suggested  that  the  field  of 
view  of  all  the  earlier  prisms  may  be  consider- 
ably enlarged  if  a  second  cut  is  made  nor- 
mal to  the  first  and  lying  above  or  even  inter- 
secting it.  The  objection  to  the  latter 
279._Bertrand  method  is  that  it  produces  a  line  across  the 
center  of  the  field.  The  angle  between  the 
cut  and  the  end  faces  is  considerably  less 
than  in  the  older  forms,  and  the  prisms,  con- 
sequently, are  shorter.  The  proportions  in  the  following  table,  which  is  in- 
serted for  comparison,  are  those  given  by  Bertrand..  They  differ  somewhat 
in  the  lengths  of  the  Nicol  and  the  Hartnack-Prazmowski  from  the  values 
given  in  Article  138. 


FIG.  278.  FIG.  279. 

FIGS,  278  AND 
prisms.  Fig.  278,  Flint  glass  with  cal- 
cite lamella;  Fig.  279,  calcite  with  glass 
lamella. 


Prisms  of  one  cut 


Prisms  of  two  cuts 


Name 

Ratio  of 
length  to 
width 

Field 
(in  air) 

Ratio  of 
length 
to  width 

Field 
(in  air) 

Nicol  prism  . 

54.2    (sic  ) 

31°  16' 

2  62 

6c°  ?4' 

Hartnack-Prazmowski  (linseed  oil  film)    . 
Calcite  prism  with  glass  plate  
Flint  glass  with  calcite  plate  . 

4.27 
4.27 
4    27 

•*    o        / 

39    34 

39    34' 
44°  2l' 

2  .02 
2  .02 
2    O2 

82°  28' 
82°  28' 
06°  30' 

Flint  glass  with  sodium  nitrate  plate  

3.416 

o        / 
52    54 

1.56 

9    o     •    i 

117  29 

129.  Ahrens'  Prisms    (1884). — Ahrens'  1884    prism1  consists  of  three 

1  C.  D.  Ahrens:    On  a  new  form  of  polarizing  prism.     Jour.  Roy.  Microc.  Soc.,  2  ser.. 
IV  (1884),  533-534- 

Idem:  On  a  new  form  of  polarizing  prism.     Phil.     Mag.,  5  ser.,  XIX  (1885),  69-70. 


ART.  131] 


THE  MICROSCOPE 


171 


FIG.  280.  PIG.  281. 

FIGS.  280  A.ND  281. — Ahrens  prisms  (1885). 


wedges  of  spar  cemented  together  by  Canada  balsam.  The  optic  axes  of  the 
two  outer  wedges  are  parallel  to  the  refracting  plane,  the  axis  of  the  middle 
one  is  perpendicular  to  it.  The  ends  are  rectangular,  and  nearly  in  contact 
with  one  of  them  is  a  prism  of  dense  glass, 
which  serves  to  deflect  one  of  the  rays 
still  farther  (Fig.  280). 

A  second  form,  in  which  the  glass 
wedge  is  cemented  to  the  calcite,  is  shown 
in  Fig.  281.  This  prism,  although  not 
having  square  ends,  seems,  on  the  whole, 
to  be  better  than  the  one  first  described. 
It  is  of  less  length,  having  a  ratio  of 
about  2  to  i,  length  to  breadth,  and  has  a 
wider  opening  angle.  A  ray  of  light, 
entering  parallel  to  the  long  axis,  is 
divided  into  two  rays,  one  of  which 
©merges  parallel  to  the  incident  ray; 
the  other  is  deflected  about  59°  30'.  The 
latter  ray  is  strongly  colored  and  distorted,  but  this  is  of  no  consequence 
since  the  deviation  is  so  great  that  it  does  not  interfere. 

130.  Madan  Prism  (iSS^.-^Madan1  suggested  that  if  a  film  of  air,  as 
in  the  Foucault  prism,  be  placed  between  two  Iceland 
spar  prisms  (a,  b,  Fig.  282),  the  ordinary  ray  will  be 
totally  reflected.  The  transmitted  extraordinary  ray, 
however,  is  deflected  and  over-corrected  for  color,  but 
both  deviation  and  dispersion  are  practically  corrected 
by  passing  them  through  a  prism  of  crown  glass  (c)  and 
one  of  very  dense  flint  glass  (d).  The  opening  angle  in 
this  prism  is  about  the  same  as  that  in  the  ordinary  nicol 
(28°)  and  much  greater  than  that  of  the  Foucault  (8°). 
While  it  is  not  quite  free  from  chromatic  aberration  and 
distortion,  this  is  not  great  enough  to  interfere  with  its 
use  as  a  polarizer.  J 

131.  Ahrens'  Prism  (1886). — Ahrens'2 1886  prism  differs 
from  most  of  those  previously  described  in  having  two 


FIG.  28  2. — M  ad  a  n 
prism. 


1  H.  G.  Madan:    On  a  modification  of  Foucaulfs  and  Ahrens's  prisms.     Nature,  XXXI 
(1884-5),  371-372. 

2  C.  D.  Ahrens:  New  polarizing  prisms.     Read  April  14,  1886.     Jour.  Roy.  Microsc. 
Soc.,  1886,  397-398. 

Silvanus  P.  Thompson:  Notes  on  some  new  polarizing  prisms.     Phil.  Mag.,  5  ser.,  XXI 
(1886),  476-478. 

Hugo  Schroder:  Ahrens'  neues Polarisations prisma.     Zeitschr.  f.  Instrum.,  VI  (1886), 


172 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  132 


section  planes  cut  through  it.  It  differs  from  the  Bertrand  prism,  and 
from  Ahrens'  1885  prism,  in  the  orientation  of  these  sections.  The 
prism  is  rectangular,  has  square  ends,  and  a  ratio  of  about  1.8  to  i 
between  the  long  and  the  short  sides.  Cry  stall  ographic  c  is  at  right 
angles  to  the  long  sides  and  passes  through  the  cut  sections  (Figs. 
283-284),  although  a  few  prisms  were  made  with  crystallographic  c 
parallel  to  the  section  plane.  The  two  oblique  cuts  meet  in  a  line  passing 
through  the  center  of  one  of  the  square  faces,  this  line  being  turned  toward 
the  source  of  light.  Cut  as  this  prism  is,  the  field  is  symmetrically  divided, 
and  the  ordinary  ray  is  reflected  to  both  sides,  leaving  an  available  polarized 
field  of  about  28°  in  one  direction  and  about  100°  in  the  other. 


Optic 


FIG.  284. 


riG.   283.  .FIG.   204. 

FIGS.  283  AND  284.  —  Ahrens  prism  (1886).     Fig.  283,  in  perspective;  Fig.  284,  in  section 


This  prism,  while  having  about  the  same  opening  angle  as  the  Nicol  or 
Thompson  prisms,  is  much  shorter.  It  has  square  ends  and  consequently  but 
little  light  is  lost  by  reflection  from  them.  There  is  very  little  distortion, 
and  it  requires  less  Iceland  spar  than  the  Nicol,  Hartnack-Prazmowski, 
Glan,  or  Thompson.  Its  chief  disadvantage  is  the  presence  of  the  section 
line  across  the  field,  although  recently  the  maker  has  cemented  a  thin  cover- 
glass  to  the  bottom  with  Canada  balsam,  thus  making  the  line  almost 
invisible.  The  prism  is  excellent  as  a  polarizer,  since  it  can  be  made  of  con- 
siderable size  with  comparatively  little  spar.  As  an  analyzer  it  is  likely  to 
produce  a  little  distortion  at  the  section  line. 

132.  Grosse  Double-slit  Air  Prism  (1890).  —  Grosse1  suggested  a  prism, 
useful  as  a  polarizer,  with  two  diagonal  intersecting  slits,  the  parts  not 
cemented,  but  separated  by  a  film  of  air  (Fig.  285). 

133.  Leiss  Prism  (1897).  —  Apparently  without  knowledge  of  Talbot's 
prism,2  Leiss3  constructed  one  on  the  same  plan,  one  half  being  made  of 
Iceland  spar  and  the  other  of  glass,  the  latter  with  a  refractive  index  as  nearly 

1  W.  Grosse:  Ueber  Polarisations  prismen.     Zeitschr.  f.  Instrum.,  X  (1890),  445-446. 

2  See  Art.  124,  supra. 

3  C.  Leiss:     Ueber  ein  neues,   aus  Kalkspath  und  Glas  zusammengesehtes  Nicol'  sches 
Prisma.     Sitzb.  Akad.  Wiss.  Berlin,  1897,  901-904. 

J.  Beckenkamp:  Review  of  above.     Zeitschr.  f.  Kryst.,  XXXIII  (1900),  112. 


ART.  135] 


THE  MICROSCOPE 


173 


as  possible  the  same  as  that  of  the  extraordinary  ray  in  the  first  half.  No 
glass  was  found  having  exactly  the  proper  index,  wherefore,  on  account  of 
the  displacement  of  the  image  on  rotating  the  prism,  it  could  be  used  only  as 
a  polarizer  and  not  as  an  analyzer. 

134.  Von  Lommel  Prism  (1898). — Independently  of  Talbot  and  Leiss, 
von  Lommel1  had  constructed,  in  1895,  a  similar  prism.  Owing  to  its  faulty 
character  he  did  not  publish  it  until  1898.  He  found  that  the  image  became 
distorted,  being  shortened  parallel  to  the  principal  section  of  the  nicol.  On 


FIG.   285. — Grosse  double  slit  air  prism.       FIG.  286. — Von  Fedorow's  polarizer.     (Fuess.) 

looking  through  it  at  lines  crossing  at  right  angles,  for  example  at  window 
bars,  they  were  found  to  appear  at  this  angle  only  when  they  were  parallel 
to  the  vibration  planes  of  the  prisms.  When  rotated  to  any  other  angle  the 
lines  crossed  at  acute  (and  obtuse)  angles. 

135.  Von  Fedorow's  Polarizer  (1901). — In  order  to  obtain  light,  plane 
polarized  as  completely  as  possible,  for  use  with  his  rotating  apparatus,  von 
Fedorow2  constructed  a  polarizing  prism  built  on  entirely  new  lines.  Instead 
of  using  a  doubly  refracting  crystal  cut  on  a  plane,  he  made  use  of  a  hemi- 
sphere of  calcite  (C,  Fig.  286)  cut  with  the  optic  axis  parallel  to  the  flat  sur- 
face, and  set  in  a  hemispherical  recess  in  a  piece  of  flint  glass  (G)  whose  refrac- 


1  E.  von  Lommel:  Ueber  aus  Kalkspath  und  Glas  zusammengesetzte  Nicol' sche  Prismen. 
Sitzb.  Akad.  Wiss.  Miinchen,  XXVIII  (1898),  111-116. 

P.  Groth:  Review  of  above.    Zeitschr.  f.  Kryst,  XXXIII  (1900),  489-490. 

2  E.  von  Fedorow:    Article  in  Russian  with  a  French  resume  in  Annuaire  geol.  et  miner, 
d.  Russie,  IV  (1900),  142-149.     Reviewed  by  V.  von  Worobieff:  Einige  Hiilfsapparate.  fur 
das  Polarisationsmikroskop.     Zeitschr.  f.  Kryst.,  XXXVII  (1902-3),  413-414. 

Idem:  Article  in  Ibidem,  V  (1902),  217-221.  Reviewed  by  P.  Groth:  Optlsche  Vorrich- 
iitngcn,  die  auf  dcr  Anii'cnditng  der  Glasplattchenpackcte  beruhen:  Zeitschr.  f.  Kryst.,  XL 
11904-5),  297-298. 

P.  Groth:  Physikalische  Krystallo graphic.      4te  Aufl.,  Leipzig,  1905,  768. 


174 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  136 


tive  index  lies  between  that  of  the  two  rays  of  the  calcite.  The  extraordinary 
rays,  passing  into  a  glass  of  higher  index,  are  strongly  refracted,  and  are 
absorbed  by  the  black  enclosing  ring.  The  ordinary  rays,  passing  into  a 
medium  of  little  less  density,  change  their  direction  but  slightly.  3  To  prevent 
the  central  extraordinary  ray  from  passing  through  the  apparatus,  a  small 
black  plate  m  is  cemented  over  the  center  to  a  glass  plate  a. 

As  first  described,  the  instrument  was  available  only  for  use  with  mono- 
chromatic light.     To  make  it  available  for  white  light  also,  there  are  placed 


FIG.  287. 


FIG.  288. 


FIGS.   287  AND  288. — Halle  prisms. 


beneath  it  two  bundles  of  thin  glass  plates  (/  and  II)  of  the  thickness  of  cover- 
glasses,  ground  on  the  flat  surfaces  and  cemented  together,  the  balsam  making 
a  film  from  0.5  to  0.75  mm.  between  each,  llhe  emerging  light,  when  it 
reaches  the  calcite  hemisphere,  is  already  nearly  plane  parallel,  and  the 
finally  emerging  ray  has  a  divergence  of  no  more  than  i°.J 

136.  Halle  Prisms  (1908). — Halle1  designed  two  modified  nicol  prisms 
giving  opening  angles  respectively  of  17°  to  19°  (Fig.  287),  and  25°  (Fig.  288). 
From  126  c.c.  of  calcite,  a  prism  of  the  first  form  37  mm.  by  67  mm.  could  be 
cut,  or  one  of  the  second  22  mm.  by  60  mm. 

137.  Glass  Polarizing  Prisms. — Stolze,2  in  1895,  described  a  polarizing 


1  Bernhard  Halle:  Ueber  Polaris ationsprismen.      Deutsche Mechan.  Zeitung,  1908,6-7, 
16-19. 

2  Stolze.     Atelier  d.  Photographen,  1895,  140.* 


ART.  138] 


THE  MICROSCOPE 


175 


prism  made  entirely  of  glass.  The  angles  of  tj,he  faces  FE  and  BC  (Fig.  289) 
are  so  chosen  that  the  ray  of  light  which  enters  and  leaves  the  prism  perpen- 
dicularly to  AB  and  ED,  is  totally  polarized  by  the  face  BC,  the  face  FE 
being  silvered  to  prevent  loss  of  light  by  reflection. 

Owing  to  the  lateral  displacement  of  the  polarized  ray,  and  to  its  incom- 
plete polarization  if  the  glass  is  strained,  this  prism  has  been  little  used.  A 
better  form,  proposed  by  Schulz,1  is  shown  in  Fig.  290.  The  emerging  ray, 
being  polarized  outside  the  glass,  is  not  affected  by  strain  in  the  glass,  nor  is 
there  any  displacement  of  the  light  ray.  While  the  intensity  of  the  emerging 


/  1 

^^/*T1 

/   1". 

/   1 

B 

FIG.   289- — Stolze  glass  polarizing  prism. 


FIG.  290. — Schulz  glass  polarizing  prism. 


light  is  but  10  per  cent,  of  that  entering,  and  in  a  nicol  prism  it  is  from  25 
per  cent,  to  40  per  cent.,  this  is  no  great  disadvantage,  since  a  glass  prism  may 
be  made  of  any  desired  size. 

138.  Summary  of  Properties  of  Polarizing  Prisms. — The  various  polariz- 
ing prisms  described  above  are  compared  in  the  following  table,  taken,  in 
part,  from  Fuessner.2 


Name 

Vibration  di- 
rection of  ray 
passing  through 
in  relation  to 
the  separating 
film 

Approxi- 
mate open- 
ing angle 

Inclination 
of  cut  plane 
to   vertical 
axis  of  prism 

Inclination 
-   of  balsam 
film  to  +he 
c  axis 

Ratio  of 
length  to 
breadth 

Nicol  
Foucault  
Hartnack-Piazmowski  
Hartnack,  oil  film  
Glan 

N* 
N 

N 

P 
P 

N 
N 
N 
N 
N 
N 
N 
X 
X 
X 

29° 
8° 
35° 

42° 

7°  56.  5' 
35° 

27° 

- 
£ 

3°o 
20° 

44°  23' 
28°  X  100° 

22° 

40° 
13°  54' 
13°  54 
50°  1  8' 
22° 

22o° 

13°    12' 

17°  24' 
20     18' 

£42' 

27° 
13°  15' 
13°  30' 
1  6° 

41°  44' 
5°  37' 
90° 
90° 

0° 
0° 
94°  16' 
90° 
90° 
9O 
90° 
90° 
90° 
0° 
90° 
74° 

3.28 
i  .528 
3-51 
4.04 
0.831 
3  .  28  and  up. 
2.5 
4.26 
3-19 
2.70 
3-53 
2.25 
1.96 
4-27 
4-27 
1-75 

Thompson  
Thompson  reversed  nicol  
Fuessner,  calcite  plate  
Fuessner,  calcite  plate  
Fuessner,  calcite  plate  

Fuessner,  sodium  nitrate  
Fuessner,  sodium  nitrate  
Fuessner,  sodium  nitrate  
Bertrand,  calcite  plate  
Bertrand,  glass  plate*  
Ahrens  (1886)  

*X  =  normal  to  the  balsam,  air,  etc.,  film. 
P  =  parallel  to  the  film. 

1  H.   Schulz:    Polarisations prismen  aus  Glas.     Zeitschr.    f.   Instrum.,   XXXI    (191 1), 
180-182. 

2  K.  Fuessner:    U?ber  die  Prismen  zur  Polarisation  des  Lichtes.     Zeitschr.  f.  Instrum., 
IV  (1884),  49- 


176 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  139 


Grosse1  gives  the  following  comparative  table  of  the  values  of  various 
prisms.  The  numbers  i  to  5  indicate  the  value  of  the  prism  in  regard  to  the 
purpose  specified  in  the  first  column.  The  last  column  gives  the  most  advan- 
tageous forms  for  each  of  these  purposes,  namely,  those  given  a  score  of  4  or  5. 


Nicol 

Air 

Double 
slit 

•S 

•M 

group 

prisms 

prisms     1  ^ 

.axj  T3 

rt  3  v 

V3  f3 

•5  °  8 

Most 

\4 

| 

3*1 

advantageous 

o 

-*-j 

tj 

<U    Q, 

form 

"o 
o 

c 

c 

A 

£        .2     S 

3 

efl 
§ 

rt 

z 

g 

XI 

3 

III 

z 

ffi 

H 

2 

5  3 

o 
fc 

u 

ffl 

i 

p    JS0^ 

i.  Plane  polarized  field. 

3 

4 

s 

2 

2 

3 

3(0          2 

i 

T.   H. 

2.  Field  of  view  

3 

3 

3             2 

2      I 

T 

4 

4 

I 

5 

Plate,  B.  Ah. 

3.  Loss  of  light  
4.  Displacement  of  ray. 
5.  Ratio    of    length    to 

4 

2 
I 

5 

S 

i 

Si         5        !   3'    2 
5          5(t)  '    5     3 
I          3         j    3     4 

2 

I 

4 

3 
3 

I 

S 
S 
3 

2 
3 

5 

I 

4 

D.  H.  T.  Ah. 
H.  T.  Ah.  (D).Ab  . 
G.  F.  Air  (double 

breadth 

slit). 

6.  Waste  of  material  .  .  . 

4(3)      2 

i!        4        '424 

5         D.  Ab.  Air,  Plate,  F. 

Total  points  i6(  +  1)  20 

20  i?(  +  4)    19  14   ,14 

17 

2l(  +  2)        I?             18        j 

I                        i 

1 

139.  Polarizer  and  Analyzer. — In  a  petrographic  microscope  there  are 
two  polarizing  prisms,  generally  called  nicols,  although  one  usually  is,  and 
both  may  be,  prisms  of  a  different  type. 

The  lower  "nicol"  is  placed  below  the  stage,  and  is  called  the  polarizer. 
Any  prism  described  above  may  be  used  as  a  polarizer  since  the  effect  of 
displacement  of  light  or  of  cross-cutting  lines  does  not  reach  the  eye. 

The  fittings  by  which  polarizers  are  attached  to  the  sub-stages  are 
different  in  different  microscopes.  Some  have  been  described  above,2  and 
are  shown  in  Figs.  255-257.  Others  may  be  seen  in  Figs.  307-324.  The 
fittings  should  be  so  made  that  the  nicol  may  be  easily  rotated,  and  yet 
fall  into  its  position  at  o°  accurately.  The  amount  of  rotation  should  be 
determinable,  the  nicol  should  be  readily  removable,  and  it  should  be  so 
arranged  that  it  can  be  elevated  or  depressed. 

The  analyzer,  in  most  recent  microscopes  (A,  Fig.  231;  N,  Figs.  310,  311, 
3iia,  etc.)  is  made  to  slide  in  a  slot  in  the  tube,  and  may  be  rotated  through 
90°  (Figs.  309,  3iia,  316),  the  amount  of  rotation  being  indicated  on  a  scale. 
Analyzers  generally  are  flat-ended  prisms,  so  made  to  avoid  displacing  the 
image.  For  ordinary  purposes  this  analyzer  is  sufficient,  but  when  certain 
accessories,  to  be  described  later,  are  used,  it  is  necessary  to  use  a  nicol  which 
fits  over  the  eyepiece,  and  which  is  called  a  cap  nicol  (Az,  Fig.  309).  Cap 
nicols  only  were  used  as  analyzers  in  the  older  types  of  microscopes,  and  their 
removal  and  replacement  was  a  most  awkward  proceeding.  With  their  use, 
too,  the  field  of  view  of  the  microscope  is  much  reduced. 

The  insertion  of  the  analyzer  in  the  path  of  the  rays  causes  a  displacement 
of  the  focus  and  it  is  necessary,  consequently,  that  there  should  be  attached 

1  W.  Grosser  Op.  cit. 

2  Article  118,  supra. 


ART.  139] 


THE  MICROSCOPE 


111 


above  it,  in  the  analyzer  carriage,  a  long  focus  lens  so  made  that  it  is  not 
necessary  to  re-focus  when  the  analyzer  is  inserted.  This  lens  must  be  very 
accurately  adapted  to  the  nicol  used,  for  the  image  must  be  perfectly  sharp 
under  all  conditions.  A  very  slight  change  in  focus  is  extremely  tiring  to 
the  eye. 

For 'certain  purposes,  such  as  measuring  small  extinction  angles,  deter- 
mining weak  pleochroism,  locating  the  points  of  emergence  of  the  optic  axes 
and  so  on,  it  is  desirable  to  rotate  both  nicols  at 
the  same  time  instead  of  rotating  the  stage.  This 
is  made  possible,  in  certain  microscopes,  by  attach- 
ing the  nicols  to  geared  wheels1  (Figs.  310,  320 
and  322),  or  by  connecting  them  by  a  rigid  bar 
(Figs.  291,  312  and  313).  The  latter  method  was 
first  used  by  Dick2  in  1888  in  his  own  instrument, 
but  when  put  on  the  market  by  Swift  &  Son,  the 
rotation  was  produced  by  geared  wheels.3  The 
first  illustration  of  a  microscope  with  rigid  bar 
connection  appears  to  be  that  of  de  Souza-Bran- 
dao,4  in  1903.  This  microscope  was  put  on  the 
market  later,  as  the  Fuess  Ib  (Fig.  312),  and  is  de- 
scribed below.5  In  this  microscope  the  connection 
between  the  nicols  is  a  simple  rigid  bar  so  ar- 
ranged that  it  may  be  detached  and  the  upper  nicol 
moved  in  or  out  independently,  as  in  the  usual  mi- 
croscopes. A  year  later  Sommerfeldt6  described 
and  illustrated  a  microscope  of  the  same  type 
which  he  used  in  high  temperature  work.  The 
analyzer,  in  this  case,  was  a  cap  nicol,  and  the 
connecting  bar,  telescopic.  In  1905  he7  described 


FIG.  291. — Sommerfeldt's 
microscope  with  simultane- 
ously rotating  nicols. 


1  C.  Leiss:    Ueber  neuere  Instrumente  und  V orrichtungen  fur  petrographische  und  krystal- 
lographische  Untersuchungen.     Neues  Jahrb.  B.B.,  X  (1895-6),  412-420. 

Idem:  Ueber  N  euconstructionen  von  Instrument  en  fiir  krystallographische  und   petro- 
graphische Untersuchungen.     Ibidem,  179-183. 

2  Allan  Dick:    A  new  form  of  microscope.     Mineralog.  Mag.,  VIII  (1888),  160-163. 

3  Anon:    Dick  and  Swift's  patent  petrological  microscope.     Jour.  Roy.  Microsc.  Soc., 
1889,  432-436. 

Anon:  Messrs.  Swift  and  Son's  improved  Dick  petrological  microscope.     Ibidem,  1895, 

97- 

4  V.  de  Souza-Brandao:     O  novo  microscopio  da  commissao  do  sen i$o  geologico.     Com- 
municadSes  da  Commissao  do  Servico  Geologico  de  Portugal,  V  (1903-1904),  118-250. 

5  See  Art.  172,  infra. 

6  Ernst  Sommerfeldt:    Ein  fiir  miner alogisc he   Untersuchungen  bei  hotter   Temper atur 
gccignetes  Mikroskop.     Zeitschr.  f.  wiss.  Mikrosk.,  XXI  (1904),  181-185. 

7  Idem:    Die  mikroskopische  Achsenwinkel  bestimmung  bti  sehr  kleinen    Kristallpra- 
paraten.     Ibidem,  XXII  (1905),  356-362. 

12 


178  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  140 

a  similar  bar  connection  (Fig.  291)  for  observing  interference  figures  in  very 
small  minerals.  Here,  however,  the  polarizer  was  connected  with  an  ana- 
lyzer in  the  inner  tube,  and  the  amount  of  rotation  was  read  from  the 
graduated  circle  of  the  stage.  The  same  principle  was  used  by  Wright1  in 
1910.  The  method  of  connecting  or  disconnecting  the  bar  differs  from  the 
preceding  in  being  by  means  of  a  hinged  fork  instead  of  a  telescopic  rod. 
This  microscope  is  illustrated  in  Fig.  313. 

After  continued  usage,  lost  motion  is  more  likely  to  develop  with  the  geared 
wheel  than  with  the  rigid  bar  connection.  The  latter,  too,  is  less  clumsy. 

140.  Determination  of  the  Vibration  Directions  in  the  Nicol  Prisms. — 
It  is  important,  for  some  determinations,  to  know  the  vibration  directions  of 
the  nicol  prisms;  for  example  in  the  determination  of  the  directions  of  ab- 
sorption in  a  crystal.     If  the  form  of  polarizing  prism  is  known,  its  vibration 
direction  may  be  found  from  the  table  in  Article  138,  the  separating  plane 
being  easily  seen  upon  looking  through  the  prism  at  an  angle.     If  the  kind  of 
prism  is  unknown,  its  vibration  direction  may  be  determined  by  examining 
plates  of  certain  minerals  through  it.     For  example,  a  section  of  biotite, 
cut  at  right  angles  to  its  cleavage,  has  its  greatest  absorption,  consequently 
it  is  darkest,  when  its  cleavage  direction  is  parallel  to  the  plane  of  vibration 
of  the  polarizer.     Tourmaline,  on  the  other  hand,  extinguishes   vibrations 
at  right  angles  to  the  optic  axis,  that  is,  it  absorbs  the  ordinary  ray,  and  only 
the  light  rays  vibrating  parallel  to  crystallographic  c  emerge.     It  is,  therefore, 
dark  when  the  elongation  (c)  is  at  right  angles  to  the  vibration  plane  of  the 
polarizer. 

Another  method  is  to  remove  the  polarizer  from  the  microscope  and 
examine,  through  it,  light  reflected  by  a  horizontal  polished  surface,  such  as 
a  plate  of  glass  or  a  varnished  table  top.  Knowing  that  light  is  polarized  in 
a  plane  parallel  to  the  reflecting  surface,2  it  is  clear  that  the  polarizing  plane 
of  the  " nicol"  lies  at  right  angles  to  the  reflecting  surface  when  the  latter 
appears  dark. 

141.  Bertrand  Lens. — In  the  upper  part  of  the  tube  of  most  modern  mi- 
croscopes, there  is  a  slot  for  the  insertion  of  a  Bertrand  lens  which,  pref- 
erably, should  be  fastened  in  a  sliding  carrier  and  permanently  attached  to 
the  tube  (BL,  Fig.  230;  B,  Fig.  309;  g,  Fig.  3na;  B,  Fig.  313,  etc.).     This 
lens,  in  connection  with  the  ocular,  acts  as  a  small  microscope  and  magnifies 

1  Fred.  Eugene  Wright:    A  new  petrographic  microscope.     Amer.  Jour.  Sci.,  XXIX 
(1910),  407-426. 

Idem:  The  methods  of  petrographic-microsco pit  research.  Carnegie  Publication  No.  158, 
Washington,  1911,  58. 

C.  Leiss:  Mikroskop  mil  gemeinsamer  Nicoldrehung  in  vereinfachter  Form.  Zeitschr.  f. 
Kryst.,  XLVII  (1909),  377~378. 

2  Art.  42,  supra. 


ART.  141]  THE  MICROSCOPE  179 

the  interference  figure  to  be  described  later.  In  order  to  bring  the  image  to 
a  sharp  focus  with  different  oculars,  the  Bertrand  lens  must  be  fastened  in 
a  sliding  collar,  ordinarily  held  in  place  by  friction,  though  in  some  micro- 
scopes it  is  moved  by  means  of  a  rack  and  pinion. 

The  Bertrand  lens  will  be  discussed  in  greater  detail  in  connection  with 
the  observation  of  interference  figures.  The  method  of  its  adjustment  in  the 
tube  of  the  microscope  is  given  in  Article  201. 


CHAPTER  X 
THE  MICROSCOPE  (Continued) 

THE  OBJECTIVE 

142.  Introductory. — The  objective  is  the  lower  lens  system  of  a  micro- 
scope, and  is  the  one  that  first  receives  the  light  from  the  object  itself.     It 
brings  the  light  to  a  focus  .in  the  principal  focal  plane  of  the  ocular  (F2,  Fig. 
229)  and  produces  there  a  real  and  magnified  image.     A  great  deal  of  care 
and  skill  are  necessary  in  making  an  objective,  and  one  of  high  power  is, 
consequently,  the  most  expensive  part  of  the  microscope. 

Objectives  may  be  classified  according  to  their  magnification.     Roughly 
one  may  consider  an  objective  as  of  low  power  when  its  focal  length  is  above 

13  mm.  and  its  magnification  (^)less  than  15  diameters;  of  medium  power 

when  its  focal  length  is  between  12  and  5  mm.  and  its  magnification  up  to 
40  diameters;  and  of  high  power  when  its  focal  length  is  less  than  4.5  mm. 
and  its  magnification  over  40  diameters. 

143.  Definition. — The  greater  the  correction  for  chromatic  and  spherical 
aberration,  the  greater  the  definition.     Owing  to  the  impossibility  of  entirely 
correcting  all  aberration,  every  point  of  the  object  will  be  represented  in  the 
image,  not  by  a  point,  but  by  a  small  circle  of  aberration,  the  size  of  the  cir- 
cles depending  upon  the  type  of  objective.     While  this  feature  is  one  that 
must  be  taken  into  account  in  biologic  work,  and  caution  taken  to  distinguish 
between  phantom  structures,  which  do  not  exist  in  the  object,  and  real  struc- 
tures, yet  it  is  one  that  may  be  practically  disregarded  in  working  wil  h  the 
comparatively  low  powers  used  in  petrographic  work. 

144.  Depth  of  Definition  (Depth  of  Focus),  or  Penetration.1 — With  low- 
power  objectives  it  is  quite  possible  to  see  sharply,  at  the  same  time,  objects 
lying  in  slightly  different  planes.    With  high-power  lenses  this  is  not  possible 
to  so  great  an  extent,  for  the  depth  of  focus  diminishes  inversely  as  the  numer- 

1  Anon:  Penetration    of  wide-angled  objectives.     Jour.  Roy.  Microsc.  Soc.,  II  (1879), 
322-323. 

George  E.  Blackham:  Penetration.     A  contribution  to  the  study  of  the  subject.     Amer. 
Jour.  Microsc.,  V  (1880),  145-150. 

E.  Abbe:  Conditions  of  micro  stereoscopic  vision.    Penetration.     Jour.  Roy.  Miscrosc. 
Soc.,  Ill  (1880),  207. 

C.  M.  Vorce:  Penetration  in  objectives — is  it  a  defect  or  an  advantage.     Amer.  Mon. 
Microsc.  Jour.,  I  (1880),  170-171. 

Anon:  Penetrating  power  of  objectives.     Jour.  Roy.  Microsc.  Soc.,   N.   S.,  I   (1881), 
831-832. 

Edward  M.  Nelson:  The  penetrating  power  of  the  microscope.     Ibidem,  1892,  331-341. 

180 


ART.  147]  THE  OBJECTIVE  181 

ical  aperture.  A  thin  section  may  be  considered  as  being  made  up  of  a  series 
of  superimposed  planes,  only  one  of  which  may  be  seen  for  one  adjustment 
of  focus.  The  penetrating  powers  for  various  values  of  numerical  aperture 
are  given  in  the  table  in  Article  1 54. 

145.  Flatness  of  Field. — An  objective  is  said  to  have  a  flat  field  when  all 
portions  of  a  flat  object  observed  through  it  appear  equally  sharp  at  the  same 
time.     Usually  the  image  around  the  edges  of  the  field  appears  blurred,  and  it 
is  necessary,  with  high  powers,  to  change  the  focus  between  center  and  sides. 
This  indistinctness  is  due  to  what  is  known  as  the  curvature  of  the  image, 
that  is,  the  image  produced  does  not  lie  in  a  true  plane  but  on  a  more  or  less 
curved  surface.     It  has  not  yet  been  possible  to  correct  this  entirely  in  high- 
power  objectives,  consequently  one  showing  such  blurring  is  not  necessarily 
to  be  condemned  as  defective. 

146.  Illuminating  Power. — The    illuminating  power    or   brightness  of 
image,  for  a  given  magnification,  other  things  being  equal,  increases  as  the 
square  of  the  numerical  aperture. 

147.  Resolving  Power. — The  resolving  power  of  an  objective  is  that 
property  by  virtue  of  which  one  is  enabled  to  see  the  finer  details  of  an  object. 
It  is  a  fixed  quantity  of  an  objective,  not  necessarily  increased  with  its  mag- 
nifying power,  but  depending  upon  the  numerical  aperture  and  the  correction 
for  spherical  and  chromatic  aberration.1    The  resolving  power  increases  with 
the  number  and  obliquity  of  the  rays  coming  from  the  object,  that  is,  it 
increases   directly   as  the  numerical  aperture,  consequently  an  immersion 
fluid,  by  increasing  the  number  of  rays  brought  to  the  object,  increases  it. 
All  lenses  having  the  same  numerical  aperture,  however,  may  not  have  the 
same  resolving  power. 

Lenses  may  be  able  to  resolve  an  object  into  details  too  fine  to  be  seen  by 
the  eye,  which  can  only  distinguish  about  250  lines  to  the  inch.  That  there  is 
a  limit  to  the  smallness  of  an  object  which  one  can  see  is  due  to  the  fact  that 
the  nerve  fibers  of  the  eye  have  a  definite  size,  and  when  the  angle  in  the  eye, 
formed  by  the  rays  from  two  points,  is  smaller  than  about  thirty  seconds  of 
arc,  these  points  appear  as  one.  Expressed  in  distances,  it  may  be  said  that 
when  two  points  are  removed  from  the  eye  6876  times  the  distance 
separating  them,  they  will  appear  as  a  single  point.  Thus  pleurosigma 
angulatum,  with  about  50,000  lines  to  the  inch,  can  be  resolved  by  a  1/2 
in.  objective  so  as  to  be  clearly  seen  with  a  3/4  in.  ocular,  but  not  with  a 
i  1/2  in.  A  much  smaller  line  may  be  seen  than  an  interval  between  two 

1  See  Edward  M.  Nelson:  On  the  limits  of  resolving  power  for  the  microscope  and  telescope. 
Jour.  Roy.  Microsc.  Soc.,  1906,  521-531. 

Sir  A.  E.  Wright:  Principles  of  Microscopy,  1906,  231. 
See  also  Article  154,  infra. 


182  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  148 

lines.  Jurin1  could  see  a  single  pin  when  the  angle  at  the  eye  was  between 
two  and  three  minutes  of  arc,  but  the  interval  between  two  pins  disappeared 
when  the  angle  was  thirty  minutes. 

148.  Working  Distance.— The  working  distance2  of  an  objective  is  the 
distance  between  its  front  lens  and  an  object  in  focus  without  cover-glass. 
The  actual  working  distance  is  lessened  by  the  cover-glass  and,  usually,  by  a 
rim  of  metal  projecting  beyond  the  lens. 

149.  Magnifying  Power. — The  magnifying  power  of  an  objective,  called 
its  initial  magnification,  depends  upon  the  tube  length  of  the  microscope.   It 
can  be  increased  by  extending  the  draw  tube.     It  should  be  measured  from 
the  equivalent  plane  of  the  objective  to  the  equivalent  plane  of  the  ocular, 
but  since  this  varies  with  different  oculars,  the  magnifying  power  of  objectives 
is  not  always  given  by  the  makers.     It  is  usually  easier  to  measure  the  mag- 
nification of  a  combination  of  objective  and  ocular  direct.     According  to 

Abbe  the  power  of  an  objective  is  obtained  by  the  formula  ~J~,3  where  F0 

r o 

is  the  focal  length  of  the  objective.  This  is  the  value  used  by  Zeiss,  Reichert, 
Swift,  and  others,  while  Bausch  &  Lomb,  Leitz,  and  others  use  ~-,  where  A  is 

f  o 

the  optical  tube  length.  These  values  are  shown,  for  various  objectives, 
in  the  table  in  Article  153. 

From  the  same  table  the  increase  in  the  magnifying  power \.p\  of  an 

objective,  produced  by  extending  the  draw  tube,  may  be  computed.  For  ex- 
ample, a  Leitz  No.  5  objective  used  with  a  mechanical  tube  length  of  180  mm. 
magnifies  33.3  times.  Increasing  the  tube  length  by  50  mm.  increases  the 
optical  tube  length  practically  the  same  amount,  and  the  value  becomes, 

N= —  =  37-9  instead  of  N  =  — -. 

1  Dr.  Royston-Pigott :  A  further  inquiry  into  the  limits  of  microscopic  vision  and  the 
delushe  application  of  Fraunhofer's  optical  law  of  vision.     Jour.  Roy.  Microsc.  Soc.,  1879, 
9-20. 

2  Ernst  Grundlach:  Working  distance  and  its  relations  to  focal   length   and    aperture. 
Amer.  Mon.  Microsc.  Jour.,  II  (1881),  32-33. 

8  Article  98,  supra. 

See  also  E.  M.  Nelson:  Virtual  images  and  initial  magnifying  power.  Jour.  Roy. 
Microsc.  Soc.,  1892,  180-185. 

Malassez  pointed  out  that  there  is  no  precise  definition  of  the  phrase  magnifying 
power.  He  suggested  that  it  be  expressed  in  terms  of  unit  distance  from  the  posterior 
face  of  the  lens.  L.  Malassez:  Eialuation  du  pouvoir  grossissant  des  objectifs  microsco piques. 
Archives  d'  Anatomic  Microsc.,  1904,  274,  285.  Comptes  Rendus,  CXLI  (1905),  1004- 
1006. 

Idem:  Sur  le  pouvoir  grossissant  des  objectifs  microscopiques.  Comptes  Rendus,  CXLI 
(1905),  880-881. 

Idem:  Evaluation  de  la  puissance  des  objectifs  microscopiques.  Ibidem,  CXLII  (1906), 
773-775- 


ART.  150] 


THE  OBJECTIVE 


183 


150.  Dry  and  Immersion  Objectives. — Objectives  may  be  classified  as 
dry  or  immersion,  depending  upon  the  medium  between  them  and  the  object. 
In  the  former,  air  only  is  used;  in  the  latter,  water,  glycerine,  oil,  or  some 
other  fluid  having  about  the  same  index  of  refraction  as  the  glass,  and  the 
lenses  are  spoken  of  as  water-,  oil-,  etc.,  immersion  objectives.  In  petro- 
graphic  work  no  great  magnifying  powers  are  required,  and  immersion  lenses 
are  not  much  used  although  they  could  be  used  to  advantage  in  certain  cases. 
Immersion  lenses  possess  the  advantage  of  transmitting  more  light,1  since 
none  is  lost  by  partial  reflection,  but  dry  lenses  gain  nothing  by  being  used 
with  oil,  since  they  were  not  constructed  with  that  object  in  view. 

The  effect  of  the  im- 
mersion oil  upon  the  re- 
solving power  of  the  ob- 
jective may  be  seen  from 
Fig.  292,  which  shows  a 
section  through  the  front 
lens  of  the  objective,  the 
cover-glass,  the  Canada 
balsam,  the  object  slide, 
and  the  condenser.  The 
left  half  of  the  lens  may 
be  taken  to  represent  the 
front  lens  of  a  Leitz  1/12- 
in.  oil-immersion,  and  the 
right  half,  that  of  a  Leitz 
No.  9  dry  objective,  the 

two  |  being  chosen  since  they  have  very  similar  front  lenses  and  work- 
ing distances.  Corresponding  to  the  immersion  oil  shown  below  the  objec- 
tive to  the  left,  there  is  an  oil  film  below  the  object  slide  and  above  the 
condenser,  shown  in  the  lower  right  half.  This  is  necessary  only  for  objec- 
tives having  a  numerical  aperture  greater  than  i.o.  If  the  refractive  indices 
of  the  condenser,  immersion  fluid,  object  slide?  mounting  medium,  and  object 
lens  are  approximately  equal,  the  ray  H  suffers  no  refraction  in  passing  from 
H  to  L,  and  the  angle  at  which  it  leaves  the  object  is  the  same  as  the  angular 
aperture,  that  is, 

a=fr  (i) 

Such  a  system  is  spoken  of  as  one  of  homogeneous  immersion. 

In  the  dry  system,  shown  in  the  upper  right  and  lower  left  halves  of  the 
figure,  the  path  of  the  ray  is  entirely  different.  The  extreme  ray  P,  corre- 
sponding to  the  ray  H  of  the  oil-immersion  lens,  is  totally  reflected  at  Q,  and 


FIG.   292. — Passage  of   light   through  dry  and  oil-immersion  ob- 
jectives.    (Leitz.) 


1  W.  H.  Dallinger:  Dry  v.  immersion  objectives.     Jour.  Roy.  Microsc.  Soc.,  I  (1878), 


154- 


184 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  150 


only  those  rays  whose  angles  of  incidence  are  less  than  the  critical  angle  can 
pass  through.  The  ray  DEFG,  for  example,  passing  from  a  denser  to  a  rarer 
medium,  follows  the  law 

sinr  =  _i  .  . 

sin  5     n 

In  the  figure,  for  comparison,  the  angular  apertures  in  the  dry  and  im- 
mersion lenses  have  been  taken  to  be  the  same,  so  that  fi  =  d  =  60°.  But  from 
equation  (i)  13  =  a,  therefore  d  =  a  and  equation  (2)  becomes 


siny      i 
sina      ri 


(3) 


That  is,  the  angular  aperture  being  the  same,  the  resolving  power  of  the  dry 
lens  (which  depends  upon  the  obliquity  and  number  of  rays  coming  from  the 

object  (Art.  147))  is  only  —  that  of  the  oil  immersion. 

The  immersion  fluid  used  is  generally  cedar  oil  (Juniperus  virginiana).     It 
should  be  kept  in  a  well-corked  bottle  so  that  it  may  remain  free  from  dust, 

and  not  thicken  from  evaporation. 
A  bottle  having  a  rod  of  glass  or 
wire  attached  to  the  stopper  (Figs. 
293-294 *)  is  very  convenient,  since  it 
affords  a  means  by  which  the  oil  may 
be  applied  to  the  objective,  and  also 
prevents  its  waste  or  accidental 
spilling. 

Oil  may  be  placed  between  the  ob- 
ject and  the  objective  in  one  of  two 
ways.  A  drop  may  be  placed  on 
the  surface  of  the  cover-glass  and  the 
tube  racked  down  until  the  objec- 
tive dips  in  the  oil,  or  it  may  be 

applied  to  the  lens  of  the  objective,  which  is  then  lowered  until  the  oil 
touches  the  cover.  The  latter  method  is  probably  the  better,  since  there 
is  less  danger  of  spreading  the  oil.  Air  bubbles  should  be  avoided.  If 
any  occurs,  the  oil  should  be  removed  and  another  drop  applied.  After 
using  an  oil-immersion  objective  both  the  lens  and  the  slide  should  be  care- 
fully cleaned  .at  once.  The  greater  part  of  the  oil  should  be  taken  up 
with  filter  paper  or  soft  linen,  and  the  front  lens  and  thin  section  wiped 
dry  with  very  soft  linen,  moistened,  if  necessary,  with  a  drop  of  xylene 
or  benzene,  never  alcohol.  Care  must  be  taken  not  to  use  too  much  benzene 
for  it  may  penetrate  between  the  lenses  and  destroy  the  balsam  film. 

JW.    Gebhardt:  Fliischen   zur   Aufbewahrung   des   Immersionsols.     Zeitschr.    f.    wiss. 
Mikrosk.,  XIV  (1897),  348-350. 


FIG.  293.— Im- 
mersion oil  bottle. 
(Bausch  and  Lomb.) 


FIG. 


294. — Immersion  oil 
bottle.      (Zeiss.) 


ART.  152] 


THE  OBJECTIVE 


185 


151.  Classification  of  Objectives  According  to  Correction  for  Aberration. 

— Another  method  of  classifying  objectives  is  according  to  the  amount  of 
correction  for  aberration,  and  objectives  may  be  achromatic,  semi-apochro- 
matic,  or  apochromatic.  The  former  are  corrected  for  primary  spherical 
aberration  and  aberration  for  one  color,  the  semi-apochromatics  are  corrected, 
in  addition,  for  a  second  color,  and  the  apochromats  are  chromatically  cor- 
rected for  three  colors  and  spherically  corrected  for  two  (See  Art.  93.) 

Three  types  of  achromatic  objectives  are  made  by  Leitz  and  are  shown  in 
Figs.  295  to  297.  They  may  be  taken  as  representative  of  the  objectives 
made  by  all  opticians.  The  first  (Fig.  295)  consists  of  two  members,  each 


FIG.   295. 


FIG.   296. 
FIGS.  295  TO  297. — Achromatic  objectives. 


FIG. 


(Leitz.) 


of  which  is  made  up  of  two  or  three  cemented  lenses,  and  is  representative,  of 
low-  and  medium-power  dry  objectives.  The  second,  a  high-power  dry  objec- 
tive, has  a  hemispherical  front  lens — the  magnifying  element — back  of  which 
there  are  two  other  members,  each  made  up  of  two  or  three  cemented  lenses 
for  correcting  the  spherical  and  chromatic  aberration.  Fig.  297  shows  a 
i/i2-in.  oil-immersion  objective.  The  front  hemispherical  lens  is  succeeded 
by  a  meniscus,  back  of  which  the  third  and  fourth  members  of  two  or  three 
cemented  lenses  each  act  as  correctors  of  aberration. 

152.  Effect  of  Cover-glasses  of  Different  Thicknesses. — The  effect  of 
cover-glasses  of  different  thicknesses  upon  the  passage  of  light  from  Canada 
balsam  to  air  is  shown  in  Fig.  298.  The  rays  of  light,  coming  from  the  object 
0,  are  refracted  away  from  the  normal  N  when  they  emerge  in  air.  Traced 
backward,  it  will  be  seen  that  the  various  rays,  even  with  the  same  thick- 
ness of  cover-glass,  do  not  intersect  in  a  point,  but  lie  at  various  distances 
from  the  objective  along  the  normal,  and  that  with  cover-glasses  of  different 
thicknesses,  the  vertical  intercepts  are  different.1  The  intercepts  represent 

1  See  also  Art.  208,  infra. 

M.  D.  Ewell  made  various  experiments  to  test  the  effect  of  curved  cover-glasses  and 
found  that  in  those  purchased  of  good  dealers  it  "was  practically  nil.  M.  D.  Ewell:  The 
effect  of  curvature  of  cover- glass  upon  microscopy.  Proc.  Amer.  Microsc.  Soc.,  i3th  meeting, 
Detroit.  XII  (1891),  79-93. 


186 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  152 


the  positions  of  the  image  for  the  various  rays,  and,  by  comparison  with 

Fig.  205,  it  may  be  seen  that  while  objectives  might  be  spherically  undercor- 

rected  so  as  to  compensate  this  effect  for  one  thickness  of  cover-glass,  it 

would  not  answer  for  some  other.     This  effect  of  cover-glasses  of  different 

thicknesses  is  not  noticeable  in  low-power  objectives,  but  a  variation  of  even 

0.05  mm.  may  greatly  impair  the  efficiency  of  the  higher  powers.     The  above 
,  remarks   do   not,  of   course,  apply  to 

homogeneous-immersion  objectives, 
and  only  in  a  small  degree  to  water- 
immersion  objectives. 

Dry,  non-adjustable  objectives  are 
generally  corrected  to  compensate  for 
cover-glasses  0.16  to  0.18  mm.  in  thick- 
ness.1 If  the  covers  are  of  a  different 
thickness,  this  may  be  partially  counter- 
balanced by  slightly  lengthening  the 
body  tube  for  those  that  are  thin,  and 
shortening  it  for  those  that  are  thick. 
Certain  high-power  objectives  are  made 
with  a  correction  collar  (Fig.  299)  by 
means  of  which  an  adjustment  may  be 
made  for  cover-glasses  of  different 
thicknesses.  By  rotating  the  collar, 
the  two  posterior  lenses  separate,  more 

or  less,   from   the  anterior.     The  divisions  on  the  scale  indicate,  in  hun- 

dredths   of  a   millimeter,   trie  cover-glass  thickness  corresponding  to  that 

adjustment. 

To  compensate  an  objective  requires  some  practice. 

It  is  best  accomplished  by  using  a  test  plate  of  some 

kind,  such  as  a  slide  of  pleurosigma  angulatum  or  an 

Abbe  test  plate.     In  the  former  a  flat  diatom  is  selected 

and  examined  for  fine  lines.     If  these  are  not  seen,  the 

correction  collar   should  be   turned   sightly   with   one 

hand  while  the  micrometer  screw  of  the  microscope  is 

moved,    above    and  below    its   focus,  with  the  other. 

This  process  is  carefully  continued  until  the  fine  lines 

begin  to    appear,  which   they  will   do  lying  above  or 

below  the  plane  of  the  diatom.     The  fine  lines  are  now 

kept  in  focus  while  the  correction  collar  is  turned  until 

the  lines  reach  the  plane  of  the  diatom.     The  process 


Object 


FIG.  298. — Effect  of  cover-g1asses  of  different 
thicknesses. 


FIG.  2  9  9. — Objec- 
tive fitted  with  correc- 
tion-collar. (Zeiss) . 


1  Bausch  &  Lomb  0.18  mm.,  Beck  0.006  in.,  Fuess  0.17  mm.,  Leitz  0.16-0.18  mm., 
Nachet  0.15  mm.,  Reichert  0.17  mm.,  Seibert  0.15-0.18  mm.,  Swift  0.18  mm.,  Zeiss 
0.17  mm. 


ART.  152] 


THE  OBJECTIVE 


187 


should  be  repeated  several  times,  and  the  mean  of  the  readings  thus  obtained 
should  be  noted  on  the  slide  as  the  proper  correction  for  that  slide  with  that 
particular  objective  and  ocular. 

The  Abbe  test  plate1  (Fig.  300),  in  its  present  form,  consists  of  a  long 
thin  wedge  of  glass,  silvered  and  engraved  with  a  series  of  parallel  lines  on  its 
under  side,  and  cemented  to  an  object  slip.  Owing  to  the  wedge-shape  of  the 
cover-glass,  there  is  a  gradual  increase  in  the  thickness  from  one  end  to  the 
other,  which  can  be  read  directly  to  o.oi  mm.,  or  less,  from  a  scale  engraved 
upon  it.  The  edges  of  the  engraving  upon  the  silver,  which  is  extremely 
thin,  serve  as  the  marks  upon  which  to  focus. 


0.10 


0.20 


Abie's 
Test-Platte 


FIG.  300.— The  Abbe  test  plate  No.  24.     (Zeiss.) 

If  one  prepares  his  own  thin  sections,  it  is  advisable  to  use  cover-glasses 
of  uniform  thickness.  Commercially  they  come  assorted  in  sizes : 

No.  i.  Thickness,  o.  13  to  o.  16  mm. 

2.  Thickness,  o.  16  to  o.  25  mm. 

3.  Thickness,  0.25  to  0.50  mm. 

The  thicker  glasses  should  be  used  for  such  objects  as  are  to  be  examined 
only  with  low  powers,  the  thin  glasses  for  oil-immersion  objectives.  Both 
of  these,  however,  are  practically  useless  for  petrographic  work  where  a  thick- 
ness of  from  0.15  to  o.i 8  mm.  is  desirable.  To  determine  the  thickness  of 
cover-glasses,  a  micrometer  screw  (Figs.  345-346)  of  some  sort  should  be 
used,2  and  the  thickness  noted  on  the  label  of  the  slide,  so  that,  if  necessary, 
the  objective  may  be  properly  compensated.  For  objects  already  mounted, 
the  cover-glass  thickness  may  be  measured  by  the  method  of  the  Due  de 
Chaulnes,  given  in  Article  208.  Another  method  is  given  by  Czapski.3 
Since  the  index  of  refraction  of  the  cover-glass  is  generally  unknown,  it  is  not 
sufficient,  in  making  this  test,  to  focus  upon  the  top  and  bottom  and  multiply 

1  E.  Abbe:    Beitrage  zur  Theorie  des  Mikroskops  und  der  mikroskopische  Wahrnehmung> 
Archiv.  f.  mikr.  Anat.,  IX  (1873),  434~437-     Also  in  Gesammelte  Abhandlungen,  I,  1904, 
66-68. 

Anon:  Abbe's  test-plate.     Jour.  Roy.  Microsc.  Soc.,  Ill  (1883),  281-283. 
Anon:  Directions  for  using  the  Abbe  test-plate.     Zeiss  circular,  Mikro  116. 

2  See  also   Edward   Bausch:  The  full  utilization  of  the  capacity  of  the  microscope,  and 
means  for  obtaining  the  same.     Microscope,  X  (1890),  289-296. 

3  S.  Czapski:  Die  Bestimmung  von  Deckglasdicken  an  fertigen  Praparaten.     Zeitschr.  f. 
wiss.  Mikrosk.,  V  (1888),  482-484. 


188  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  152 

the  thickness  thus  found  by  1.5  as  an  assumed  refractive  index  of  the  glass. 
In  Czapski's  method  it  is  necessary  that  cover-glasses  of  known  thicknesses, 
or  an  Abbe  test  plate,  be  used  as  a  gage.  The  objective  of  0.6  to  0.9  N.  A. 
is  focussed  on  the  top  and  bottom  of  the  known  glasses,  with  central  illumi- 
nation, and  the  amount  of  lift  of  tube  for  each  thickness  is  noted.  It  makes 
no  difference  whether  the  true  value  of  a  division  of  the  fine  adjustment  screw 
is  known  or  not.  The  values  thus  determined  are  compared  with  the  known 
thicknesses  of  the  cover-glasses,  and  a  mean  reduction  factor  is  obtained,  a 
factor  which  is  only  to  be  used  with  the  same  combination  of  objective, 
ocular,  diaphragm,  and  tube  length.  For  example,  if,  with  a  cover-glass 
0.220  mm.  in  thickness,  the  movement  of  the  micrometer  screw  was  52 

2  2 

divisions,  and  with  a  thickness  of  0.180  mm.,  43  divisions,  we  have  —  = 

o 

0.00423  and =  0.00418,  a  mean  of  0.0042,  which  is  the  factor  required.    To 

43 
make  determinations,  all  that  is  necessary  is  to  multiply  the  reading  obtained 

through  an  unknown  cover-glass  by  the  factor,  and  the  result  is  the  thickness 


NOTES  FOR  TABLE  153 

1  Data  obtained  directly  from  the  makers. 

2  Oculars  used  in  obtaining  the  field  of  view:  Bausch  &  Lomb  i  3/5  in.  (with  -^-  =  6.4), 

Fuess  No.  2  (5.6),  Leitz  No.  o  (4.0),  Seibert  No.  i  (5.0),  Zeiss  No.  2  (6.4),  Beck  No.  i  (5.0), 
Reichert  No.  II  (6.0). 

J.  W.  Stephenson:  On  a  table  of  numerical  apertures,  showing  the  equivalent  angles 
of  aperture  of  dry,  water-immersion,  and  homogeneous-immersion  objectives,  with  their  respective 
resolving  powers,  taking  the  wave  length  of  line  E  as  the  basis,  a  =  n  sin  w,  n  =  refractive  index, 
and  w  =  i/2  angle  of  aperture.  Jour.  Roy.  Microsc.  Soc.,  II  (1879),  839-841. 

Anon:  Notes  on  aperture,  microscopical  vision,  and  the  value  of  wide-angled  immersion 
objectives.  Ibidem,  N.  S.,  I  (1881),  303-360. 

Anon:  Penetrating  power  of  objectives.     Ibidem,  I  (1881),  831-832. 

NOTES  FOR  TABLE  154 

1  Frank  Crisp:  On  the  limits  of  resolution  in  the  microscope.     Ibidem,  V  (1885),  968-973. 

H.  J.  Detmers:  The  numerical  aperture  of  an  objective  in  relation  to  its  angle  of  aperture 
in  air,  water  and  balsam.  Proc.  Amer.  Microsc.  Soc.,  8th  meeting,  Cleveland,  VII  (1185), 
199-202. 

Edward  M.  Nelson:  On  the  limits  of  resolving  power  for  the  microscope  and  telescope. 
Jour.  Roy.  Microsc.  Soc.,  1906,  521-531. 


ART.  153] 


THE  OBJECTIVE 


189 


153.     COMPARATIVE  TABLE  OF  DRY  ACHROMATIC  OBJECTIVES  OF  DIFFERENT 

MAKERS1 


No. 

Maker 

F.  in  mm. 

N.  A. 

2    U 

Free 
working 
distance 

Field 
of 
view2 

Magnification 

A 
F 

250 
F 

oo 

2 
OO 
2" 
0 

oo 

ao 
i* 
i 

0 

800 

ai 
75 

&2 
0 

i  i/3* 

i 
i 

33 

aa 

i 
3 

2 

79 
80  1 
2 
3 
3 
AA 

2/3" 

802 
A 
4 
3a 

2 

B 
802A 
4 
5 
4 
3 

4- 

i 

803 

99 
101 
5 
5 
5 
7 
6 
5 
D 
6 
i/6L 
I/6S 
804 

7a 

r/, 

7 
1/8" 
805 
113 
US 
7 
8a 
9 
8 
6a 
9 

Fuess  
Nachet  
Reichert  
Bausch  &  Lomb  
Fuess  
Seibert 

6i.O 
50.0 
50.0 
48.0 
31-0 
45-0 
45-0 
42.0 
40.0 
40.0 
40.0 
39-0 
38.0 
37.0 
36.0 
32.0 
32.0 
3O.O 
28.0 
26.0 
25-4 
25.0 
24.0 
24.0 

22.  O 
22.0 

18.5 

17.0 
17.0 

16.2 
16.0 
16.0 
15.0 
14.0 
13.0 
12.7 

12.0 
12.0 
12.0 
IO.O 
10.  0 

8.5 
8.0 
8.0 
7.0 
6.4 
6.3 
6.0 
6.0 
6.0 
6.0 
5.4 

5-2 

5-0 
4-3 
4-2 
4-2 
4-0 
4-0 
4-0 
4-0 
3-5 

3-2 
3-2 

3-0 
3-0 
3-0 
30 
3-0 
3-0 
3-0 

2.8 

2.7 
2-3 
2.  I 
2.O 

O.  10 

0.09 
0.06 
0.08 

0.07 

12° 
10° 
7° 
9° 

'    '    'go'    '    ' 

70.00 
30.00 
42.00 
53-00 
39-00 
32.00 
32.00 
40.00 
34-5 
35.0 
34-0 
20.  o 
19-5 
30.0 
30.0 
38.0 
31.0 
25.0 
33-0 
14.0 
14-0 
8.0 
16.0 
14-5 
14.0 
15.0 
5.5 
ii  .0 
7.5 
5-5 
7.0 
7.5 
9.0 
8.0 
3.2 
4-0 
3-0 
7.0 
3-5 

4-2 
2.O 
2.5 
2.0 

1.6 

1.8 

2.O 
I.I 
I  .O 
I.O 

0.7 
1.25 
0.76 
0.5 
0.85 
0.40 
0.6 
0.6 
0.42 
0.6 
0.3 
0.64 
0.6o 
0.35 
0.30 
0.75 
o.  29 

0.2 
O.36 

0.4 
0.35 
0.45 
0.30 
0.60 
o.  20 
0.25 

O.2O 

1C.  00 

"&.'so' 

9.00 
4-75 
6.5 
14.0 
8.5 
7-0 
7-5 
6-5 
14.0 

's.o 

5-2 

4.8 
4-5 
5-0 
4-5 
4-0 
3-6 

3-0 

2.0 
3-2 

2-7 
3-2 

4.0 

7-5 
3-7 

5-2 

4-0 

5-0 
6.5 
8.0 
8.6 
18.0 
5.8 

4.0 
5-0 
5-0 
5-0 
8.0 
5-5 
5-5 
6.0 
6.0 
6.0 
6.0 
6.4 
6.5 
6.5 
6.5 
8.0 
8.0 
8.0 
9.0 
9-5 
IO.O 
IO.O 

10.4 
10.5 
ii.  3 
ii.  3 
13-5 
14.5 
14.5 
IS.  4 
15.6 
15.6 
16.7 
18.0 
19.0 

20.0 
20.8 
20.8 
20.8 

25.0 

25.0 
30.0 

31.0 

31.0 

36.0 
39.0 
39.6 
4L7 
4L7 
4L7 
4L7 
46.0 
48.0 
50.0 
58.0 
60.0 
6o.O 
62.5 
62.5 
62.5 
62.5 
71.0 
78.0 
78.0 
83.0 
83.0 
83.0 
83.0 
83.0 
83.0 
83.0 
89.0 
93-0 
109.0 
H9.0 
125.0 

Zeiss  
Leitz  
Leitz  
Reichert  
Beck 

0.08 

O.  II 

0.06 
0.06 

9° 

12° 

£ 

Zeiss  
Swift  
Zeiss  
Seibert  
Bausch  &  Lomb  
Fuess  
Reichert 

o.  20 

O.II 
O.  IO 
0.  10 

o.  17 


23° 

•  •  -I2°'  '  ' 

11°   29' 

11° 

19° 

Zeiss  
Zeiss  
Seibert 

0.17 

0.22 
0.22 

o.  19 
0.25 
0.13 
0.21 

0.34 

O.  21 
0.30 
O.3O 

0.25 
0.15 
o.  20 
0.26 
0.40 
0.26 

0.35 
0.30 
0.30 

0.32 
0.47 
0.35 
o.  50 
0.50 
0.40 
0.60 
0.80 
0.68 
0.80 
0.88 
o  60 
0.68 
0.77 
0.82 
0.85 
o.  85 
0.65 
0.82 
0.65 
0.85 
0.71 
0.90 
0.88 
0.85 
0.82 
o.  85 

0.85 
0.82 
0.92 

0.97 
O.9O 
0.90 
0.97 
0.90 
0.90 
0.95 

19.6° 

25l 
25° 

24o 
29° 

I5o 

r; 
if 

Ifs- 

17° 
23-2° 

30° 
47° 
30° 

4Io 

35° 
40° 
38° 
56° 
40° 
60° 
60° 
47° 
74° 
1  06° 
85° 
1  06° 
123° 
80° 

101° 
100° 
110° 

116° 
128° 
81.2° 

110° 
8i°4' 
116°  26' 
90° 
120° 
122° 
128° 
110° 

116° 
116°  26' 

110° 
134° 
145° 
125° 
128° 

152° 

130° 
128° 
142° 

Nachet  
Leitz  
Swift  
Beck  
Fuess  
Reichert  •  . 

4.0 

3-35 
2.8 
2.  2 
1-9 

2.5 

.  I 

.85 

.  2 
.O 

.6 
.6 

.  2 
'.I 

.  2 
.  I 

•  5 
-45 
0.9 
0.9 
I.O 
0.7 
0.8 

0.70 
0.6 
0.55 
0.5 
o.S 
0.5 
0.48 
0.43 
0.43 
0.50 

0.40 
0.43 
0.40 
0.35 
0.32 
0.33 

6.0 

"s.s' 

Zeiss  
Leitz 

10.3 
IO.O 
9-0 

14.0 

Bausch  &  Lomb  
Beck  
Zeiss  
Fuess  
Leitz 

14.1 
14.2 

Seibert  
Zeiss  
Beck  
Xachet  
Fuess 

16.0 
27.0 

"  'l8!2' 

21.4 

20.  o 
20.  o 
31-0 

34-2 

Leitz  
Seibert 

Reichert  

Bausch  &  Lomb  
Zeiss  
Seibert  
Fuess  
Beck  
Swift 

30.0 
41.0 
4i-5 
5O.O 
33-3 
3O.O 

Swift  
Xachet  
Leitz 

Reichert  
Fuess  
Reichert  

37-0 

55-0 
48.0 
43-0 
43-0 
44-0 
80.0 
50.0 
80.0 

Seibert  
Zeiss 

Leitz  
Bausch  &  Lomb  

Beck  
Xachet  
Reichert  
Seibert  
Fuess 

Leitz  
Bausch  &  Lomb  
Beck  
Swift  
Swift  
Nachet  
Reichert  
Fuess  
Nachet  
Seibert  
Reichert 

62.5 
57-0 
60.0 
81.0 

'  '0:32' 
o.  30 

o.  28 
0.25 

IIO.O 

57-0 

130.0 
116.0 
80.0 

190 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  154 


154.    APERTURE  TABLE 


Corresponding  angle  (2  w)  for 

Limit  of  resolving  power,  in  lines 
to  an  inch 

Pene- 

Aperture 

Illumi- 

trating 

(n  sin  M  = 
N.A.) 

Air 
(n—  i.oo) 

Water 
(«=l.33) 

Homogen- 
eous im- 
mersion 
(n=  1.52) 

White  light 
line  E) 

Monochro- 
matic (blue) 
light  (A  = 
0.4861  ft, 
line  F) 

Photogra- 
phy  (A  = 
0.4000  /(, 
near  line  h 

nating 
power 
(N.A.)* 

power 
i 

N.A. 

0.05               5°  44' 

4°  18' 

3°  46' 

4,821 

5,252 

6,350 

0.003 

20.000 

0.  10 

11°  29' 

8°  38' 

7°  34' 

9,641 

10,450 

12,700 

O.OIO 

10.000 

0.15 

17°  14' 

12°  58' 

11°  19' 

14,462 

15,676 

19,050 

0.023 

6.667 

o.  20 

23°    4' 

17°  18' 

15°     7' 

19,282 

20,901 

25,400 

0.040 

5.000 

0.25 

28°  58' 

21°  40' 

18°  56' 

24,103 

26,126 

31,749 

0.063 

4.000 

0.30 

34°  56' 

26°    4' 

22°  46' 

28,923 

3L35I 

38,099 

0.090 

3-333 

0.35 

40°  58' 

30°  30' 

26°  38' 

33,744 

36,576 

44,449 

o.  123 

2.857 

0.40 

47°     9' 

35°    o' 

30°  31' 

38,564 

41,801 

50,799 

o.  1  60 

2.  5OO 

0.45 

53°  30' 

39°  33' 

34°  27' 

43,385 

47,026 

57,149 

0.203 

2.222 

0.50 

60°     o' 

44°  10' 

38°  24' 

48,205 

52,252 

63,499 

0.250 

.OOO 

0.52 

62°  40' 

46°     2' 

40°    o' 

50,133 

54,342 

66,039 

o.  270 

•923 

0.54 

65°   22' 

47°  54' 

41°  37' 

52,061 

56,432 

68,579 

0.292 

.852 

o.  56 

68°     6' 

49°  48' 

43°  14' 

53,990 

58,522 

71,119 

0.314 

.786 

0.58 

70°  54' 

51°  42' 

44°  Si' 

55,9i8 

60,612 

73,659 

0.336 

.724 

0.60 

73°  44' 

53°  38' 

46°  30' 

57,846 

62,702 

76,199 

0.360 

.667 

0.62 

76°  38' 

55°  34' 

48°     9' 

59,774 

64,792 

78,739 

0.384 

.613 

0.64 

79°  36' 

57°  31 

49°  48' 

61,702 

66,882 

81,279 

0.410 

.562 

0.66 

82°  36' 

59°  30' 

51°  28' 

63,631 

68,972 

83,819 

0.436 

•SIS 

0.68 

85°  41' 

61°  30' 

53°     9' 

65,559 

71,062 

86,359 

0.462 

•  471 

o.  70 

88°  51' 

63°  3i' 

54°  50' 

67,487 

73,152 

88,899 

0.490 

.429 

0.72 

92°  06' 

65°  32' 

56°  32' 

69,415 

75,242 

91,439 

0.518 

.389 

o.74 

95°  28' 

67°  37' 

58°  16' 

71.343 

77,333 

93,979 

0.548 

•  351 

o.  76 

98°  56' 

69°  42' 

60°    o' 

73,272 

79,423 

96,518 

0.578 

.316 

0.78 

102°  31' 

71°  49' 

61°  45' 

75.200 

8i,5i3 

99,058 

0.608 

.282 

0.80 

106°  16' 

73°  58' 

63°  31' 

77,128 

83,603 

101,598 

0.640 

.250 

0.82 

110°  10' 

76°     8' 

65°  i  8' 

79,os6 

85,693 

104,138 

0.672 

.220 

0.84 

114°  17' 

78°  20' 

67°    6' 

80,984 

87,783 

106,678 

0.706 

.  190 

0.86 

118°  38' 

80°  34' 

68°  54' 

82,913 

89,873 

109,218 

0.740 

.I63 

0.88 

123°  17' 

82°  51' 

70°  44' 

84,841 

9L963 

111,758 

0.774 

.136 

0.90 

128°  19' 

85°  10' 

72°  36' 

86,769 

94,053 

114,298 

0.810 

.  Ill 

0.92 

133°  Si' 

87°  32' 

74°  30' 

88,697 

96,143 

116,838 

0.846 

.087 

0.94 

140°    6' 

89°  56' 

76°  24' 

90,625 

98,233 

H9,378 

0.884 

.064 

.96 

147°  29' 

92°  24' 

78°  20' 

92,554 

100,323 

121,918 

0.922 

.042 

•  98 

157°     2' 

94°  56' 

80°  17' 

94,482 

102,413 

124,458 

0.960 

.020 

.00 

1  80°     0' 

97°  3i' 

82°  17' 

96,410 

104,503 

126,998 

.000 

.OOO 

.02 

100°   10' 

84°  i  8' 

98,338 

106,593 

129,538 

.040 

0.980 

.04 



102°  53' 

86°  21' 

100,266 

108,684 

132,078 

.082 

O.962 

.06 

105°  42' 

88°  27' 

102,195 

iio,774 

I34,6i8 

.124 

0.943 

.08 

108°  36' 

90°  34' 

104,123 

112,864 

137,158 

.166 

o  .  926 

.  10 

in0  36' 

92°  43' 

106,051 

114,954 

139,698 

.  2IO 

0.909 

.  12 

1  14°  44' 

94°  55' 

107,979 

117,044 

142,237 

.254 

0.893 

•  14 

118°    o' 

97°  n' 

109,907 

119,134 

144,777 

.300 

o.  877 

.16    ..;:.:;;:. 

121°  26' 

99°  29' 

111.835 

121,224 

147,317 

.346 

0.862 

.  18 

125°    3' 

101°  50' 

113,764 

123,314 

149,857 

•  392 

o  .  847 

.  20 

128°  55' 

104°  15' 

115,692 

125,404 

152,397 

.440 

0.833 

.22           

133°    4' 

106°  45' 

117,620 

127,494 

154,937 

.488 

0.820 

•  24 

137°  36' 

109°  20' 

119,548 

129,584 

157,477 

.538 

0.806 

.26 

142°  39' 

in0  59' 

121,477 

131,674 

160,017 

588 

0  .  794 

.28      !  

148°  42' 

114°  44' 

123,405 

133,764 

162,557 

.638 

/  V4- 

o.  781 

•  30        

155°  38' 

H7°  35' 

125,333 

135,854 

165,097 

.690 

0.769 

•32 

165°  56' 

120°  33' 

127,261 

137,944 

167,637 

•  742 

O.758 

•  33        

180°    o' 

122°      6' 

128,225 

138,989 

168,907 

.769 

0.752 

•34        

123°  40' 

129,189 

140,035 

170,177 

•  796 

0.746 

•  35         

125°  18' 

130,154 

141,080 

171,447 

.823 

0.741 

•  36         

126°  58' 

131,118 

142,125 

172,717 

.850 

0.735 

•  37 

128°  40' 

132,082 

143,170 

.877 

o  .  729 

•  38       ,  

130°  26' 

133,046 

144.215 

175)257    ' 

.904 

0.725 

•  39         

132°  16' 

134,010 

145,260 

176,527 

•  932 

0.719 

•  40         

134°  10' 

134,974 

146,305 

177,797 

.960 

0.714 

.41         

136°     8' 

135,938 

147,350 

179,067 

.988 

0.709 

.  42       i 

138°  12' 

136,902 

148,395 

180,337 

.  016 

o  .  704 

•  43         

140°  22' 

137,866 

149,440 

181,607 

•  045 

0.699 

•  44 

142°  39' 

138,830 

150,485 

182,877 

•  074 

o  .  694 

•45 

145°    6' 

139,795 

I5L530 

184,147 

.103 

0.690 

.46 

147°  42' 

140,759 

152,575 

185,417 

.  132 

0.685 

•  47 

ISO0  32' 

141,723 

153,620 

186,687 

.161 

0.680 

.48 

153°  39' 

142,687 

154,665 

187,957 

.  190 

0.676 

•  49 

157°  12' 

143,651 

155,7  10 

189,227 

.  220 

o  .  671 

•  SO 

161°  23' 

144,615 

156,755 

190,497 

-250 

0.667 

1.51 

166°  51' 

145,579 

I57,8oo          191,767 

.  280 

0.662 

1.52 

1  80°      0' 

146,543 

158,845          193,037 

.310 

0.658 

ART.  155] 


THE  OBJECTIVE 


191 


If  light  between  E  and  F  (  =  0.508^/1)  is  used,  the  N.  A.  will  be  a  true 
measure  of  the  resolving  power,  since  it  is  exactly  equal  to  the  number  of 
hundred  thousands  of  lines  to  an  inch.  This  will  give  100,000  as  the  maxi- 
mum for  a  dry  objective,  133,000  for  a  water-immersion,  and  153,000  for  a 
homogeneous-immersion  with  crown-glass  cover.1 

155.  Testing  the  Objective. — The  value  of  an  objective  depends  upon 
its  definition  and  resolving  power.  In  making  a  test  one  should  have  an 
objective  of  known  value  for  comparison,  and  a  series  of  test  objects.  The 
ocular  employed  should  be  the  same  in  each  case.  The  test  plate  most  com- 
monly used  is  made  by  J.  D.  M  oiler2  and  consists  of  a  slide  upon  which 
are  mounted  a  series  of  twenty  diatoms3  whose  markings  vary  from  3  to  95 
in  a  thousandth  of  an  inch.  They  are  as  follows: 


Diatom 

Direction 
of  striae 

Striae  in  i/iooo  of 
an  inch,  after 
Morley 

i    Triceratium  favus  Ehrbg 

3    7 

2    Pinnularia  nobilis  Ehrbg 

transv. 

13  o 

3    Navicula  lyra  Ehrbg   var 

transv. 

16  .0 

4    Navicula  lyra  Ehrbg       ...                          

transv. 

24.  5 

5.  Pinularia  interrupta  Sm.  var  
6    Stauroncis  phocniccnteron  Ehrbg 

transv. 

26.0 
•24.    c 

7    Grammatophora  marina  Sm  

transv. 

38.4 

8    Pleuro^igma  Balticum  Sm 

transv. 

33    I 

9    Pleurosigma  acuminatum  (Kg  )  Grun 

transv. 

46  .4 

10    Nitz'-chia  amphioxy^  Sm 

49  •  2 

1  1    Pleurosigma  angulatum   Sm                     

diagonal 

47.0 

12.  Grammatophora  oceanica  Ehrbg  =  G.  subtilissima.  . 
13.  Surirella  gemma  Ehrbg  
14.  Nitzschia  sigmoidea  Sm  
15.  Pleurosigma  fasciola  Sm.  var  
1  6    Surirella  gemrrifi  Ehrbg                                           

transv. 
transv. 
transv. 
transv. 
longit. 

61.6 

53-5 
62  .0 
58.0 
67  .0 

1  7    Cymatopleura  elliptica  Breb 

63  .0 

18.  Navicula  crassinervis  Breb  =  Frustulia  saxonica  Rabh 
10    Nitzschia  curvula  Sm                                



86.0 
90.0 

20    \mphipleura  pellucida  Kg  

transv. 

95.2 

The  process  of  testing  an  objective  serves  not  only  the  purpose  of  deter- 
mining its  limit  of  capacity,  but  teaches  a  student,  as  nothing  else  will,  how 
to  bring  out  that  capacity.  While  this  is  of  much  less  importance  in  petro- 
graphic  than  in  biologic  work,  it  is,  nevertheless,  something  that  should  be 

1  See  J.  W.  Stephenson:  Op.  cit. 

2  Anon:    Moeller's   test-plate   (Probe-Platte) .     Amer.  Jour.  Microsc.,  I   (1875),   16-17. 
Made  by  J.  D.  Moller's  Institut  fiir  Mikroskopie.     Wedel  i.  Holstein,  Germany. 

3L.  Dippel.  Zeitschr.  f.  Mikrosk.,  II  (1880),  4  plates.*  Another  test  plate  is  de- 
scribed by  Henri  Van  Heurck:  Nouielle  plaque  d'epreuve  (Test-Platte}  pour  la  verification 
des  objectifs.  Zeitschr.  f.  angew.  Mikrosk.,  IV  (1898),  1-4. 


192 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  155 


understood  by  every  user  of  a  microscope.  This  was  well  expressed  by  Hirst l 
who  said: 

"The  tyro,  sitting  down  before  his  newly  acquired  instrument,  places  an  object 
on  the  stage,  turns  on  the  full  glare  of  light  from  his  mirror  and  condenser,  and 
fancies  he  sees  everything  to  perfection.  Let  him  try  the  same  method  of  proceeding 
on  some  delicate  diatom- valve;  and  where  in  the  hand  of  the  skilled  manipulator 
a  moment  before,  lines  or  beading  were  beautifully  displayed,  he  sees  a  blank. 
He  may  spend  long  hours  in  trying  every  trick  of  illumination,  moderating  his  light, 
varying  its  obliquity  by  altering  the  angle  of  his  mirror,  focussing  and  re-focussing 
the  condenser,  altering  the  adjustment  of  his  objective;  and  at  last,  when  his  patience 
is  well-nigh  exhausted,  the  desired  result  is  obtained,  the  delicate  markings  start 
suddenly  into  view,  and  he  possesses  the  consciousness  that,  under  his  hands,  mirror, 
condenser,  and  objective  are  now  doing  their  best.  Has  this  time  been  wasted? 
I  think  not." 

The  method  of  testing,  briefly,  is  as  follows.  Place  the  objective  to  be 
tested  in  the  microscope  and  examine  all  the  diatoms  of  the  test  plate  in 
order,  beginning  with  No.  i.  At  the  start  use  the  greatest  possible  amount  of 
light,  placing  the  mirror  in  the  axial  line  of  the  microscope  and  removing  the 
polarizer.  Examine  the  structure  of  the  diatoms  and  note  whether  the  out- 
line and  the  markings  appear  to  lie  in  a  single  plane.  If  they  do  not,  adjust 
the  correction  collar  (Art.  152).  Proceed  in  the  examination  until  a  diatom 
is  reached  whose  markings  cannot  be  seen.  Now  swing  the  mirror-bar 
slightly  to  one  side,  thus  giving  more  inclined  illumination.  If  the  markings 
do  not  yet  appear,  increase  the  inclination  until  they  do.  Proceed  to  the 
next  diatom  and  so  on  until  no  striae  can  be  seen.  It  is  quite  probable  that 
by  tilting  the  mirror,  changing  the  illumination,  inserting  a  bull's-eye  con- 
denser, or  moving  the  correction  collar,  they  will  appear.  It  is  possible 
that  the  ocular  is  of  too  low  a  power.  This  may  be  determined  by  noting  how 
close  together  the  striae  were  in  the  last  diatom  in  which  they  could  be  seen. 
Successive  trials  will  probably  enable  the  student,  with  the  same  combina- 
tion of  objective  and  ocular,  to  see  striae  where  none  appeared  before. 

One  should  be  able  to  resolve  the  diatoms  given  below  by  means  of  objec- 
tives having  the  numerical  apertures  noted  in  the  first  column  of  the  table. 


N.  A. 

Diatom 

Striae  in 
o.ooi  in. 

Remarks 

o.45 

O    ere 

Pleurosigma  Balticum  
Pleurosigma  acuminatum  

33 
4.C 

Central  illumination. 
Central  illumination. 

0.65 

0-75 
0.85 

o  o< 

Pleurosigma  angulatum  
Nitzschia  sigmoidea  
Surirella  gemma  (longit.)  
Navicula  crassinervis     .    . 

47 
62 
67 
86 

Central  illumination. 
Central  illumination. 
Central  illumination. 
Inclined  illumination. 

I    OS 

Nitzschia  curvula  . 

GO 

Inclined  illumination. 

1.20 

Amphipleura  pellucida  

95 

•Inclined  illumination. 

1  G.  D.  Hirst:    Notes  on  some  local  species  of  diatomacea.     Jour,  and  Proc.  Roy.  Soc. 
New  South  Wales,  XI  (1877),  272-277,  in  particular  276. 


ART.  157] 


THE  OCULAR 


193 


156.  Cost  of  Objectives. — As  a  matter  of  comparison  it  may  be  said  that 
objectives  with  a  focal  length  of  25  mm.  and  over,  cost  approximately  $4.00 
each;  between  25  and  10  mm.,  $5.50  to  $10.00;  10  to  3  mm.,  $7.00  to  $15.00; 
3  to  2  mm.,  about  $20.00.     A  i/i2-in.  (1.9  mm.)  oil-immersion  objective 
costs  about  $27.00,  and  a  i/i6-in.,  $40.00. 

Apochromatic  objectives  are  much  more  expensive.  One  of  16  mm.  focal 
length  will  cost  about  $25.00,  8  mm.,  $32.00;  4  mm.,  $45.00;  3  mm.,  $50.00; 
2  mm.  oil-immersion  of  1.30  N.  A.,  $100.00,  and  the  same  with  1.40 N.A., 
$130.00. 

While  it  will  not  be  necessary  to  caution  owners  of  microscopes  in  regard 
to  the  care  of  their  objectives,  the  above  prices  may  serve  as  a  hint  to  students 
using  University  property.  Instructions  for  the  care  of  objectives  are  given 
in  Article  198. 

THE  OCULAR  OR  EYEPIECE 

157.  Huygens  Eyepiece. — The  ocular  of  a  microscope  is  not  nearly  so 
complicated  as  the  objective.     In  most  forms  but  two  lenses  are  used.     Three 
types   are  made,  Huygens  or  negative,  Ramsden  or 

positive,  and  compensating  eyepieces. 

The  Huygens  eyepiece  consists  of  two  simple 
plano-convex  lenses  placed  with  their  plane  surfaces 
toward  the  eye.  The  upper  lens  (e,  Fig.  301)  is 
known  as  the  eye-lens,  the  lower  (/),  as  the  collective 
or  field  lens.  The  focal  length  of  the  eye-lens  is  one- 
third  that  of  the  field-lens,  and  the  two  are  separated 
a  distance  equal  to  the  sum  of  their  focal  lengths.  The 
Huygens  eye-piece  cannot  be  used  to  magnify  an  ob- 
ject directly,  and  it  is,  for  this  reason,  called  negative. 
As  may  be  seen  from  the  figure,  the  real  image  (Os)  is 
formed  within  the  ocular  by  the  field-lens.  This  col- 
lects the  rays  which  come  from  the  objective  and 
which  would  normally  have  produced  the  real  image 
at  O2.  The  image  Oa  is  smaller  than  the  real  image 
produced  by  the  objective,  consequently  the  field  of 
view  of  the  ocular  is  greater  than  it  would  be  were  the  image  Oz  viewed 
directly.  When  cross-hairs  or  micrometers  are  used,  they  must  be  placed 
in  the  plane  of  Os  in  order  that  they  may  be  viewed  simultaneously  with  the 
image. 

The  rays  of  light  emerging  from  the  eye-lens  are  parallel,  and  thus  cause 
the  eye  least  fatigue.  Under  this  condition  the  image  appears  to  be  that  of 
an  object  infinitely  distant,  although  it  is  customary,  in  computing  magnifica- 
tions, to  consider  the  image  as  being  formed  at  the  distance  of  distinct  vision 
(250  mm.). 

13 


FIG.  301. — Huygens  or 
negative  eyepiece. 


194 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  158 


The  Huygens  eyepiece  is  the  one  most  commonly  used  in  petro- 
graphic  microscopes.  It  is  achromatic  in  the  sense  that  images  of  different 
colors  appear  of  the  same  size.  In  most  modern  instruments  the  various 
oculars  are  so  mounted  that  their  lower  focal  points  lie  in  the  same  plane  when 
inserted  in  the  tube.  That  is,  the  optical  tube  length,  except  so  far  as  this  is 
changed  by  the  ocular  itself,  remains  practically  the 
same  for  the  same  objective,  irrespective  of  the  ocular 
used.  This  is  a  great  convenience,  as  it  makes  re-focus- 
sing unnecessary  when  changing  from  one  ocular  to 
another  of  different  power. 


FIG.  302. — Ramsden  or 
positive  eyepiece. 


158.  Ramsden  Ocular. — The  Ramsden  or  positive 
ocular,  like  the  Huygens,  consists  of  two  simple  plano- 
convex lenses,  but  in  this  eyepiece  they  are  placed  with 
their  convex  sides  toward  each  other  (Fig.  302). 
Usually  the  focal  lengths  of  the  two  are  equal,  and  the 
distance  between  them  is  about  one-third  the  sum  of 
their  focal  lengths.  The  focal  plane  of  the  combination 
lies  one-fourth  the  focal  length  of  the  collective  lens/ 
below  it,  consequently  cross-hairs  or  micrometers,  placed 
in  the  focal  plane,  are  viewed  directly  through  the  eye- 
lens  e. 
This  type  of  ocular  is  used  principally  for  special  work,  such  as  making 

measurement  with  a  micrometer. 

In  neither  the  Huygens  nor  the  Ramsden  ocular  is  any  attempt  made  to 

correct  spherical  aberration,  since  they  are  used  with  small  apertures  and  the 

distortion  is  slight.     Chromatic  aberration  is  corrected  only  so  far  as  this  is 

possible  by  varying  the  distance  between  the  lenses. 

159.  Compensating    Oculars. 

— Even  in  apochromatic  objec- 
tives it  has  been  found  impossi- 
ble to  do  away  entirely  with 
differences  in  the  focal  planes  for 
different  colors.  To  overcome 
this,  Abbe  invented  compensating 
oculars.  These  are  overcorrected 
just  the  proper  amount  to  elimi- 
nate the  error,  whereby  the  field 
becomes  entirely  free  from  color  up  to  the  edge  of  the  diaphragm,  which 
itself  shows  an  orange  border.  There  is  not  a  great  deal  of  advantage  in 
using  compensating  oculars  in  petrographic  work  since  high-power  objectives 
do  not  give  perfectly  flat  fields  up  to  the  margin  where  chromatic  aberra- 
tion interferes. 


FIG.  303. — Compensating  oculars.     (Zeiss.) 


ART.  161] 


THE  OCULAR 


195 


Like  the  Huygens,  the  mounts  of  compensating  oculars  are  made  so  that 
their  lower  focal  points  fall  in  the  same  plane  (Fig.  303). 

160.  COMPARATIVE  TABLE  OF  HUYGENS  OCULARS  OF  DIFFERENT  MAKERS 


Number 

Maker 

Focal  length 
in  mm. 

Magnification 

250 
F 

A 
F 

o 

i 
i 
i 
i 

0 

i 
2" 

2 
II 

A 

i  3/5" 
rA 
II 

2 

Yif" 

ni 

B 
IV 

2 

4 

i" 

IV 

4 
V 
C 

4/5" 

2A 

V 

5 

t 

2/3" 

i 

4 

Leitz             

62.5 
57-0 
50.0 
50.0 
50.0 
50.0 
50.0 
50.0 

45-0 
41.65 
41.6 
40.0 
40.0 
40.0 
40.0 
34-0 
33-0 
31-25 
30.0 
30.0 
30.0 
30.0 

27-7 
25.0 
25.0 
25.0 
25-0 
25.0 

21  .O 
20.85 
20.8 

20.  o 
20.  o 
20.  o 

20.  O 
17.0 
I6.7 
I6.7 
I6.5 
13-8 
12-5 

4   O 

Fuess  

4*4 

5-o 
5-o 
5-0 
5-0 
5-o 
5-o 
5-6 
6.0 
6,0 
6-3 
6-3 
6-3 
6-3 
7-3 
7-9 
8.0 

8-3 
8-3 
8-3 
8-3 
9-o 

IO.O 
10.  0 
IO.O 
IO.O 
IO.O 
12  .O 
12  .O 
12.0 

12.5 

12-5 
12.5 
12.5 
14-7 
15  -0 
15-0 
15-0 

18.0 

20.0 

Leitz 

Beck 

Reichert                       ... 

3-o 
3-5 
3-0 

Seibert  

Zeiss  
Bausch  &  Lomb 

Fuess 

Leitz 

Swift 

Bausch  &  Lomb  
Beck  



Reichert  
Zeiss  
Seibert 

4.0 
5-0 

Bausch  &  Lomb 

Leitz  .  . 



Beck         

Fuess  

Reichert  
Zeiss 

5-5 

7.0 

7.0 

Swift  

Reichert 

Seibert  .  . 

Zeiss  

Bausch  &  Lomb  
Leitz  
Fuess 

Leitz  . 

Swift  .  . 



Bausch  &  Lomb  

Beck  

Reichert  
Zeiss 

IO.O 
IO.O 

Seibert  . 

Swift  

Bausch  &  Lomb  
Beck  



Swift   . 

Seibert  

14.0 

Huygens  oculars  are  worth  from  $1.50  to  $2.00;  compensating  oculars,  $6.00  to  $7.00. 

161.  Oculars  for  Special  Purposes. — Most  oculars  of  special  design  are 
used  for  observations  in  polarized  light.  They  will  be  described  below 
(Chapters  XXV-XXVI).  The  following  eyepieces  are  used  for  observa- 
tions in  ordinary  light. 


196 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  162 


162.  Demonstration  Oculars. — As  long  ago  as  1848,  Queckett1  described 
an  ocular  fitted  with  a  pointer  for  purposes  of  demonstration.  In  the  Huy- 
gens  ocular,  designed  by  Professor  Pfitzner2  and  shown  in  Fig.  304,  the  ex- 


FIG.  304. — Demonstration  ocular 
after  Pfitzner.     (Leitz.) 


PIG.  305. — Demonstration  ocular 
after   Bourguet.      (Reichert.) 


tremity  of  a  pointer,  which  is  attached  to  a  rod,  lies  in  the  plane  of  the  image. 
By  combining  a  rotation  of  the  ocular  within  the  tube  with  a  rotation  of  the 
pointer,  any  part  of  the  field  may  be  shown  to  the  student  without  centering 


FIG.  306. — Double  demonstration  ocular  after  Edinger.     (Leitz.) 

it  under  the  cross-hairs.  In  a  similar  ocular,  after  Bourguet3  (Fig.  305), 
any  mineral  may  be  pointed  out  by  inserting  the  rod,  more  or  less,  and  ro- 
tating it. 

A  double  demonstration  ocular,4  with  the  eye-lenses  separated  by  18  cm. 
(7  in.),  is  shown  in  Fig.  306.     By  means  of  reflecting  prisms,  it  is  possible 

Queckett:  Microscope,  ist  ed.,  1848,  130,  Fig.  91.* 

•Martin  Kuznitzky:  Facultative  Demonstrations-Ocular e.     Zeitschr.  f.   wiss.  Mikrosk., 
XIII  (1896),  145-146. 

3  Anon:  Neues    Index-Okular   nach    Bourguet.     Zeitschr.    f.    angew.    Mikrosk.,    VIII 

(1902),  33- 

*L.    Edinger:  Das  Zeigerdoppelokular.     Zeitschr.  f.    wiss.  Mikrosk.,   XXVII  (1910), 

336-338. 


ART.  165]  THE  OCULAR  197 

for  student  and  instructor  to  view  the  same  section  at  the  same  time.  The 
pointer,  shown  at  the  left  of  the  diagram,  may  be  pushed  in,  more  or  less,  and 
moved  in  azimuth  in  a  sliding  ring,  thus  covering  every  part  of  the  field. 

163.  Focussing  Cross-hairs  in   the   Ocular. — The   cross-hairs   of   the 
ocular  are  attached  to  a  sliding  sleeve  within  the  tube,  and  are  so  placed  that 
they  lie  in  the  plane  O3  (Figs.  301-302).     While  the  focal  plane  occupies  a 
different  position  for  different  eyes,  it  is  ordinarily  not  necessary  to  move  the 
sleeve,  the  adjustment  being  accomplished  by  sliding  the  eye-lens  collar, 
which,  in  most  oculars,  is  held  by  friction.     It  would  be  an  improvement  if 
the  eye-lens  ring  were  screwed  in  and  held  in  place  by  a  bearing  screw.     The 
easiest  way  to  adjust  the  focus  of  the  cross-hairs  is  to  remove  the  eyepiece 
from  the  microscope  and  focus  by  looking  through  it  against  a  light  back- 
ground.    The  cross-hairs  should  be  seen  in  sharp  focus  at  the  first  glance 
through  the  ocular,  and  before  the  eye  has  had  time  to  accommodate  itself. 
When  in  proper  adjustment  and  with  the  ocular  in  the  microscope,  the  cross- 
hairs will  appear  well  defined  and  lie  in  the  plane  of  the  image  of  whatever 
object  is  viewed  through  the  instrument. 

Sometimes  it  is  impossible  to  obtain  a  sharp  focus  by  shifting  the  eye- 
lens.  It  is  then  necessary  to  move  the  sliding  collar  to  which  the  cross- 
hairs are  attached.  It  is  a  simple  enough  matter  to  slide  it  into  proper  posi- 
tion in  the  Ramsden  ocular,  but  is  more  difficult  in  the  Huygens,  where  it 
lies  between  the  two  lenses.  The  eye-  or  the  field-lens  should  be  removed, 
and  the  cross-hair  collar  shoved  up  by  means  of  a  pencil,  care  being  taken 
not  to  touch  the  cobwebs.  It  may  be  necessary  to  make  several  trials  before 
getting  the  proper  position  for  the  hairs,  which  should  be  in  focus  when  the 
eye-lens  slide  is  approximately  in  its  intermediate  position. 

164.  Replacing  Cross-hairs. — The  finest  cross-hairs  are  made  of  spider 
web,  the  dark  thread  from  the  inside  of  a  nest  being  the  best.     These  nests, 
which  may  be  found  in  the  autumn  hanging  on  bushes,  should  be  torn  open 
and  the  eggs  removed,  otherwise  the  newly  hatched  spiders  will  eat  the  web. 

To  replace  cross-hairs,  first  remove  the  ring,  which  is  to  support  them, 
from  the  ocular.  It  will  be  seen  that  there  are  two  scratches,  at  right  angles 
to  each  other,  which  indicate  the  proper  positions  for  the  cross-hairs.  Take 
a  single  thread,  an  inch  or  two  long,  from  a  spider's  nest,  and  attach,  to  each 
end,  as  heavy  a  weight  as  it  will  carry.  Hold  one  weight  in  the  fingers,  and 
dip  the  thread  in  hot  water  or  hold  it  in  steam,  to  stretch  it.  Now  move  the 
ring  against  the  center  of  the  wreb  and  turn  it  into  a  horizontal  position,  leav- 
ing a  weight  to  hang  down  on  either  side.  If  the  hair  is  not  quite  in  proper 
position,  move  it  by  means  of  a  pin,  then  fasten  it  in  place  by  a  bit  of  wax 
or  a  touch  of  shellac.  Replace  the  other  hair  in  the  same  manner. 

165.  Magnification  of  the  Compound  Microscope. — According  to  Abbe, 


198  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  165 

250 
the  magnifying  power  of  an  objective  is  determined  by  the  formula  -vr- 

*  o 

(Art.  149),  and  that  of  the  eyepiece  by  ^r    (Arts.  98   and  103).      Many 

250 
makers  reverse  the  formulae  and  give  -~-  as  the  magnifying  power  of  the 

A 
eyepiece  and  ^    as  that  of  the  objective  (Art.  149).     The  magnification  of 

r  o 

the  compound  microscope  may  be  considered  as  the  resultant  of  two  succes- 
sive magnifications,  the  first  being  the  magnification  produced  by  the  objec- 
tive (03,  Fig.  229),  the  second  that  produced  by  the  ocular,  which  magnifies 
the  real  image  derived  from  the  objective  (02)  and  produces  the  final  image 
04.  Its  value,  consequently,  regardless  of  whether  Abbe's  system  or  the 
reverse  is  used,  will  be: 

250      A_250A 
"         X-~' 


But  ~=F  (Ecl-  J>  Art-  I02)> 

therefore  N  =—?,  (2) 

r 

which  is  the  same  equation  as  (2)  Art.  102,  as  it  should  be. 

In  practice1  the  magnifying  power  of  any  combination  of  ocular  and 
objective  may  be  obtained  by  direct  comparison  as  explained  in  Article  246, 
or  it  may  be  obtained  by  multiplying,  according  to  equation  i,  the  known 
magnifying  powers  of  the  ocular  and  the  objective  obtained  from  the  table  in 
Articles  153  and  160. 

If  the  tube  length  used  is  greater  than  that  given  in  the  table  of  computed 
magnifications  (160  or  170  mm.),  a  correction  must  be  made  to  the  amount 
of  magnification  of  the  objective,  as  indicated  in  Article  149.  It  is  not 
advisable,  however,  with  an  objective  of  focal  length  shorter  than  from  5  to 
7  mm.,  to  try  to  increase  the  magnification  by  changing  the  tube  length 
from  that  for  which  it  was  designed.  A  difference  of  only  10  mm.  with  an  oil- 
immersion  lens  will  materially  reduce  its  efficiency. 

1  Cf.  Sir  A.  E.  Wright:  On  certain  new  methods  of  measuring  the  magnifying  power  of 
the  microscope  and  of  its  separate  elements.  Jour.  Roy.  Microsc.  Soc.,  1904,  279-288. 


CHAPTER  XI 
VARIOUS  MODERN  MICROSCOPES 

1 66.  Introduction. — It  is  impracticable  to  describe  all  of  the  different 
kinds  of  petrographic  microscopes  made,  and  in  the  following  pages  only 
some  of  the  more  important  instruments  will  be  noted.     While  the  stands 
described  below  are  typical  of  those  of  the  different  makers,  there  are  in- 
numerable varieties,  especially  of  more  simplified  form,  and  the  catalogues 
of  the  different  manufacturers1  may  be  examined  with  profit  by  the  student 
as  a  supplement  to  this  chapter. 

167.  Leitz    Stand    AM. — One  of    the  best  petrographic  microscopes 
manufactured  is  the  Leitz 's  stand  AM,2  already  described  in  part  and  repre- 
sented in  Figs.  230  and  231.    The  stand  is  of  large  dimensions,  without  being 
clumsy,  and  provides  ample  space  for  all  of  the  accessories  used  in  modern 
petrographic  work,  including  v.  Fedorow's  universal  stage.    The  body  tube 
is  unusually  wide  so  that  it  may  be  used  in  photomicrographic  work.    The 
draw  tube  (TA,  Fig.   231)  is  adjustable  by  means  of  a  rack  and  pinion  moved 
by  the  milled  head  OcE,  and  is  graduated  to  show  the  mechanical  tube  length. 
The  fine  adjustment  has  been  described  above  (Art.  115).    The  revolving 
stage  may  be  read  to  minutes  by  means  of  a  vernier,  and  may  be  moved  slowly 
by  means  of  a  tangent  screw  (TS,  Fig.  230),  the  latter  movement  being  ex- 
tremely valuable  in  reading  small  extinction  angles  and  so  on.    The  stage 
also  has  two  lateral  movements  with  a  range  of  20  mm.,  the  motion  being 
controlled  by  two  screws,  and  its  amount  read  by  means  of  two  scales  set  into 
the  stage. 

The  polarizer  (P,  Fig.  231)  and  iris  diaphragm,  which  are  shown  in  detail 
in  Figs.  258-260,  may  be  raised  or  lowered  by  means  of  the  screw  BT.  Both 

1  Some  of  the  leading  manufacturers  of  petrographic  microscopes  are: 
America:  Bausch  &  Lomb  Optical  Co.,  Rochester,  N.  Y. 

Austria:  C.  Reichert,  Bennogasse  24-26,  Wien  VIII. 

England:  R.  &  J.  Beck,  Ltd.,  68  Cornhill,  London.  James  Swift  &  Son,  81  Tottenham 
Court  Road,  London,  W. 

France:  A.  Nachet,  17  Rue  St.  Severin,  Paris. 

Germany:  R.  Fuess,  Steglitzb.  Berlin.  DuntherstrasseS.  E.  Leitz,  Wetzlar.  (Branch, 
30  East  1 8th  St.,  New  York.)  W.  &  H.  Seibert,  Wetzlar.  Carl  Zeiss,  Jena. 

Switzerland:  Societe  Genevoise  pour  la  Construction  d'Instruments  de  Physique  et 
de  Mechanique,  8  Rue  des  Vieux-Grenadiers,  Geneve. 

2  Gabriel  Lincio:  Das  neue  Leitz' sche  mineralogische  Mikroskopmodell  A.    Neues  Jahrb., 
B.  B.,  XXIII  (1907),  163-186. 

199 


200 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  168 


'sXW 


polarizer  and  analyzer  are  of  the  Glan-Thompson  type  with  a  large  opening 
angle.  They  may  be  rotated  and  the  amount  of  rotation  read  from  a  divided 
scale.  Above  the  analyzer  is  a  long  focus  lens  to  correct  the  displacement 
caused  by  the  insertion  of  the  prism.  The  Bertrand  lens  (BL,  Fig.  230) 
slides  in  and  out  of  the  draw  tube,  and  may  be  moved  up  or  down  to  bring 
it  into  focus. 

The  only  improvements 
that  might  be  suggested  for 
this  stand  are  the  addition  of 
a  diaphragm,  either  sliding  or 

f      BK!        *r*S>  *n  ^e  image  plane  of  the 
Bertrand  lens,   and  a  detach- 
Mjjjj^Bff  able,  rigid  connection  between 

the  two  nicols,  permitting  their 
simultaneous  rotation,  as  in 
the  instruments  shown  in  Figs. 
291,  312,  and  313. 

168.  Leitz's  Berkey  Model. 

— A  simpler  microscope,  and 
one  most  excellently  adapted 
to  the  use  of  elementary 
students,  is  shown  in  Fig.  307. 
It  was  made  after  the  specifi- 
cations of  Professor  Berkey 
and  embraces  the  most  essen- 
tial accessories.  Being  less 
elaborate  than  the  preceding, 
it  is  less  likely  to  get  out  of 
order,  a  considerable  advantage 
in  general  class-room  work.  In 

-^jgjjjmjjjj^m      KJ?     this  microscope  the  condensing 
flHI  ^BBHIi  Pi     lens  is  inserted  or  thrown  aside 

by  rotating  the  milled  head 
beneath  the  stage.  The  polar- 
izer may  be  placed  in  the  o°, 
90°,  1 80°  or  270°  positions,  or  may  be  swung  entirely  aside.  The  revolv- 
ing stage  is  graduated,  the  analyzer  slides  in  and  out  of  the  tube,  and  the 
Bertrand  lens  may  be  slipped  into  a  slot  above  it.  The  graduated  draw- 
tube  has  an  inside  diameter  of  24  mm. 

i68a.  Leitz's  New  Stand. — A  microscope,  somewhat  more  complete 
than  the  above  and  embracing  all  the  essentials  of  a  modern  instrument 
and  at  a  reasonable  price,  is  shown  in  its  preliminary  form  in  Fig.  308. 


FIG.  307. — Berkey  model  microscope.     (Leitz.) 


ART.  168a] 


VARIOUS  MODERN  MICROSCOPES 


201 


Instead  of  the  usual  tube,  24  mm.  in  diameter,  this  instrument  has  a  tube  of 
30  mm.,  and  thus  has  a  field  of  view  nearly  twice  as  great.  Both  coarse  and 
fine  adjustments  are  provided.  The  latter  is  of  the  type  shown  in  Fig.  245, 
and  has  divisions  of  o.oi  mm.,  permitting  a  reading  to  0.0025  mm«  with  ease. 
There  are  two  iris  diaphragms,  one  above  the  lower  nicol,  and  one  above 
the  Bertrand  lens.  The  lower  diaphragm  and  the  whole  condensing  system 
may  be  raised  or  lowered  by  means  of  a  rack  and  pinion,  and  the  diaphragm 
may  be  displaced  laterally  so  that 
inclined  illumination  may  be  used. 
The  Bertrand  lens  is  fastened  in  a 
slider  in  the  inner  tube  whereby  it 
may  be  raised  or  lowered  to  bring  it 
into  focus  with  any  ocular.  It  has 
two  adjusting  screws  at  the  side  for 
accurate  centering.  The  upper 
lenses  of  the  condensing  system  may 
be  thrown  aside  by  rotating  a  milled 
head,  and  are  so  constructed  that  in 
changing  from  parallel  to  convergent 
light,  it  is  not  necessary  to  lower  the 
polarizer.  The  angular  aperture  of 
the  condenser  is  large,  giving  the  first 
yellow  ring  around  the  interference 
figure  of  quartz.  The  Johannsen 
wedge,  inserted  in  a  slider  in  the  ac- 
cessory slot  at  45°,  makes  unneces- 
sary the  picking  up  and  laying  down 
of  mica  plate  or  quartz  wedge  for 
each  determination.  Attached  to 
each  objective  is  a  centering  device 
which  consists  of  two  screws,  work- 
ing at  right  angles  to  each  other,  and 
adjusted  by  means  of  watch  keys. 


FIG.  308. — New  petrographic  microscope.     (Leitz.) 


When  this  is  combined  with  the  new 
objective  tongs,  in  which  a  strong 
spring  presses  firmly  against  an  in- 
clined  bar,   permanent  centering  is  obtained.     An  attachable  mechanical 
stage  may  be  used  with  the  instrument  if  desired. 

In  a  later  instrument  made  for  the  writer,  the  adjusting  screws  for  the 
Bertrand  lens,  shown  in  the  illustration,  have  been  replaced  by  two  square- 
end  screws  which  may  be  turned  by  means  of  the  same  watch  keys  used  in 
centering  the  cross-hairs.  This  prevents  the  accidental  displacement  of 
the  screws  by  the  finger.  The  Bertrand  lens,  also,  is  inserted  from  the  other 


202 


MANUAL  OF  PETROGRAPUIC  METHODS 


[ART.  169 


side  from  that  shown  in  the  figure,  to  correspond  in  position  to  that  of  the 
analyzer.  This  arrangement  places  the  levers  for  both  iris  diaphragms  to 
the  rear  when  they  are  open,  instead  of  one  to  the  front  and  one  to  the  rear. 
There  is  under  consideration,  furthermore,  a  new  device  for  the  simulta- 
neous rotation  of  the  nicols. 

169.  Seibert  Microscope. — The  Seibert 
microscope  shown  in  Fig.  309  is  another 
excellent  instrument.     It  has  coarse  and 
fine  adjustment  screws,  one  division  on 
the  latter  measuring  0.002  mm.;  revolv- 
ing stage,  provided  with  degree  divisions, 
vernier,  and  stage  clamp;  and  centering 
screws  working  parallel  to  the  cross-hairs, 
for  adjusting  the  center.     Both  polarizer 
and   analyzer   are  flat-end   nicols.     The 
former  may  be  moved  up  or  down  by 
means  of  a  screw,  and  may  be  swung  aside 
by  means  of  a  hinge  when  observations  are 
to  be  made  by  ordinary  light.     Between 
the  polarizer  and  the  condenser  is  an  iris 
diaphragm.     The  Bertrand  lens  is  inserted 
by  raising  the  lever  just  above  the  ana- 
lyzer.    As  in  many  microscopes,  there  is 
no  upper  diaphragm,  although  for  modern 
petrographic  work  it  is  almost  absolutely 
necessary. 

170.  Fuess  Stand  Via. — One  of   the 
newest  of  the  Fuess  microscopes  (Via) l  is 
shown  in  Fig.  310.     It  differs  from  those 
previously  described  in  having  an  attach- 
ment by  means  of  which  the  nicols  may  be 
simultaneously  revolved.     In  older  forms 
the  rotating   analyzer  was    a   cap   nicol 

3/7  which  materially  cut  down  the  field  of 
view.  In  this  microscope  the  tube  is 
double,  the  outer  one  being  stationary  while  the  inner  one  rotates,  carrying 
with  it  the  nicol  prism  and  the  slot  above  the  objective  clip  (k).  It  thus  per- 
mits any  accessory  placed  in  this  slot  to  retain  its  orientation,  with  reference 
to  the  nicols,  during  the  rotation.  It  is  possible,  also,  to  rotate  (i)  polarizer, 

1  J.  Hirschwald:  Ueber  ein  neues  Mikroskopmodell,  etc.     Centralbl.  f.  Min.,  etc.,  1904, 
626-633. 

Idem:  Die  Priifung  der  naturlichen  Bausteine  auf  Ihre  Weiterbesiandigkeit.     Berlin. 
Idem:  Handbuch  der  bautechnischen  Gesteinspriifung.     Berlin  I,  1911,  142-147. 


FIG.  309. — Petrographic  microscope, 
natural  size.     (Seibert.) 


ART.  171] 


VARIOUS  MODERN  MICROSCOPES 


203 


analyzer,  and  ocular  with  cross-hairs  simultaneously,  (2)  analyzer  and  ocular, 
(3)  polarizer  and  analyzer,  or  (4)  analyzer  alone.  The  polarizer  is  a  modi- 
fied nicol,  the  analyzer  a  Glan-Thompson.  In  the  eyepiece  there  is  a  sliding 
diaphragm  containing  a  circular  and  a  square  opening,  the  latter  to  facilitate 
the  measuring  of  all  of  the  constituents  of  a  rock  section.  The  stage  is  of 
the  Hirschwald  pattern,  described  above.1  The  graduations  of  the  stage 
are  in  degrees,  with  a  vernier  read- 
ing to  5'.  A  novel  arrangement  is 
the  electric  light  (G)  for  illuminat- 
ing opaque  minerals  or  rock  slabs, 
a  blue  glass  in  front  of  the  light  re- 
ducing its  yellow  color  to  approxi- 
mately the  tone  of  daylight. 

171.  Fuess  Stand,  Ilia. — One  of 

the  best  moderate-priced  instruments 
for  students'  use  is  the  Fuess  Ilia.2 
Fig.  311  represents  the  stand  of  the 
No.  Ill  which  becomes  Ilia  by  the 
substitution  of  the  tube  shown  in 
Fig.  3iia.  The  following  descrip- 
tion applies  to  the  Ilia. 

The  rotating  stage  is  divided  into 
degrees,  with  verniers  reading  to  5'. 
The  polarizer  is  a  modified  nicol 
prism  with  square  cross-section  and 
cemented  with  linseed  oil.  It  is 
raised  or  lowered  by  means  of  a 
milled  head  on  the  side  of  the  in- 
strument not  shown  in  the  illustra- 
tion. The  upper  lenses  of  the  con- 
densing system  may  be  thrown  in 
or  out  by  means  of  the  lever  bf,  also 
shown  in  Fig.  233.  The  objective 
clip  is  shown  in  Fig.  239.  The  fine 
adjustment  is  produced  by  means  of 
a  spring  depressed  by  the  milled  head  n,  which  is  graduated  and  acts  as  a 
micrometer  screw. 

The  particularly  attractive  feature  of  this  microscope  is  its  extremely 
wide  field  of  view.     The  field,  ordinarily,  is  limited  by  the  size  of  the  collec- 

1  Article  109,  supra. 

2  C.  Leiss:  Mikroskope  mil  sehr  grossem  Sehfeld  fur  petrographische  Studien.     Neues 
Jahrb.,  II  (1897),  86-88. 


?0 


FIG.  310. — Microscope  model  VI  a.  (Fuess.) 


204 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  171 


tive  lens  of  the  ocular.  In  this  microscope  the  tube  is  30  mm.  in  diameter 
instead  of  23.25  mm.  as  in  most  microscopes,  and  by  this  means  it  is  pos- 
sible to  use  larger  oculars.  The  following  table  gives,  in  millimeters,  the 


FIG.   3iia. — Large   tube  for  micro- 
scope III  a.     (Fuess.) 


field  of  the  new  and  of  the 
older  forms,  and  shows  that 
it  has  been  approximately 
doubled. 

The  analyzer  is  a  Glan- 
Thompson  prism  which  may 
be  rotated  by  means  of  the 
lever  d.  The  Bertrand  lens, 
in  the  newer  instruments,  is 
permanently  attached  at  /, 
and  beneath  it  is  an  iris 
diaphragm  /.  A  diaphragm 
above  the  polarizer  (Fig. 
255)  should  be  specified  in 
ordering. 


PIG.  311. — Microscope  model  III. 
(Fuess.) 


1/3  natural  size. 


ART.  172] 


VARIOUS  MODERN  MICROSCOPES 


205 


Fuess'  objective  number 

0               I 

2 

3 

4 

5 

6 

7 

8 

9 

Fuess'  ocular  usual  No.  2  . 

3-8     3-45 

2.25 

1.6 

i-35 

0.9 

0.7 

0.46 

o-33 

0.28 

Fuess'    large   field   ocular, 

6.0     5.5 

3-31 

2-5 

2.0 

i-S 

O-iS 

0.7 

0-55 

0.4 

Xo.  2. 

It  would  be  an  improvement  if  the  condensing  lens  and  iris  diaphragm 
were  attached  to  a  holder  separate  from  the  polarizer,  so  that  the  latter  might 
be  swung  aside  without  the  former.  This  might  be  so  arranged  that  one 
could  displace  the  center  of  the  condenser  and  the  diaphragm  from  the  axis 
of  the  microscope  for  the  production  of  inclined  illumination.  The  iris 
diaphragm  lever  (/,  Fig.  3iia),  shown  in  the  illustration  as  straight,  is  usually 
made  with  a  drop.  It  should  be  straight,  for  although  not  quite  so  easy  of 
access,  it  does  not  continually  twist  and  strain  the  threads  as  does  the  bent 
form. 

172.  Fuess  Microscope,  Model  Ib. — On  a  previous  page1  there  was  il- 
lustrated a  rigid  bar  connection,  after  designs  by  Sommerfeldt,  by  which 
the  two  nicols  could  be  simultaneously  rotated.  A  similar  arrangement, 
on  a  more  complicated  instrument,  is  shown  in  Fig.  312.  This  microscope, 
after  de  Souza-Brandao,2  consists  of  a  large  stand,  similar  to  the  Fuess  No. 
la,  and,  like  that  instrument,  has  an  Abbe3  illuminating  apparatus.  The 
polarizer  is  an  Ahrens  prism,  and  over  it  are  the  condensers,  which  are  centered 
by  means  of  the  screws  z.  The  stage  possesses,  besides  the  usual  rotation 
in  azimuth,  a  second  rotation  in  altitude,  the  amount  being  read,  by  means 
of  verniers  and  the  drum  7\,  to  5  minutes.  This  movement  is  very  con- 
venient, especially  for  obtaining  maximum  extinction  angles,  which  is  usually 
possible  since  the  opening  in  the  stage  is  6  cm.  in  diameter  on  the  lower  side, 
and  the  stage  may  be  rotated  as  much  as  60°.  The  mechanical  stage  mm\ 
has  micrometer  divisions  to  o.oi  mm.  and  is  detachable,  a  very  good  point 
since  for  most  purposes  it  is  more  convenient  to  work  without  it.  The  tube, 
which  cannot  be  lengthened,  has  an  inside  diameter  of  24  mm.,  and  takes 
the  ordinary  oculars.  The  analyzer  is  a  Glan-Thompson  prism  which  may 
be  rotated  independently  through  135°  when  the  bar  connecting  it  with  the 
polarizer  is  placed  in  the  90°  position.  The  simultaneous  rotation  of  the 

1  Article  139  and  Fig.  291,  supra. 

-  \ .  de  Souza-Brandao:  O  novo  microscopio  da.  commissao  do  semi^o  geologico  Com- 
municadoes  da  Commissao  do  Service  Geologico  de  Portugal,  V  (1903-4),  118-250. 

C.  Leiss:  Ueber  zu'ei  neue  Mikroskope  fiir  petrographische  und  krystalloptische  Studien. 
Zeitschr.  f.  Kryst.,  XLIX  (1910-1911),  193-197. 

3  C.  Leiss:  Ncucs  Mikroskopmodell  la  fiir  mineralogische  und  petrographische  Studien. 
Zeitschr.  f.  Kryst.,  XLIV  (1908),  264-267. 

Idem:  Neucs  Mikroskop  Modell  VIb  fiir  krystallographische  und  petrographische 
Studien.  Ibidem,  XL VIII  (1910),  240-242. 


206 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  172 


nicols  is  produced  by  means  of  the  bar  s,  which  is  attached  to  the  divided  circle 
N,  the  connection  with  the  nicols  being  made  by  means  of  the  forked  bars 
J3  and  s±.  The  amount  of  rotation  possible  is  180°,  the  vernier  reading  to 
5  minutes.  Below  the  ocular  is  the  iris  diaphragm  OJ.  Accompanying  the 


FIG.  312. — De  Souza-Brandao  microscope.     (Model  I  6.)     (Puess.) 

microscope  are  three  Bertrand  lenses  adapted  for  different  oculars,  thus  re- 
quiring their  removal  and  insertion,  a  method  less  convenient  than  a  per- 
manently attached  Bertrand  lens  set  in  a  sliding  sleeve,  as  in  the  Fuess  Ilia 
microscope,  and  adapted  for  all  oculars.  Below  the  stage  is  an  iris  diaphragm  /. 


ART.  173] 


VARIOUS  MODERN  MICROSCOPES 


207 


173.   Fuess     Micro- 
scope, Model  Ha.— The 

latest  microscope  with  si- 
multaneously rotating 
nicols  is  the  Fuess  Model 
Ha1  (Fig.  313).  As  may 
be  seen  from  the  illustra- 
tion, a  rigid  bar  connects 
hinged  levers  extending 
from  polarizer  to  analy- 
zer, the  object  of  the 
hinges  being  to  permit 
the  end  portions  to  be 
elevated  and  thus  allow 
the  nicols  to  be  slipped  in 
or  out,  or  rotated  inde- 
pendently. The  amount 
of  rotation  of  the  nicols 
may  be  read  from  the 
graduated  circle  above 
the  analyzer  or  from  the 
graduations  of  the  stage. 
The  analyzer  is  a  Glan- 
Thompson  prism,  the 
polarizer  an  Ahrens.  If 
the  rotating  lever  of  the 
upper  nicol  were  attached 
beneath  the  calcite  prism, 
it  would  be  advantageous 
since  it  would  do  away 
with  the  reflection  of 
light  from  its  upper  sur- 
face. In  this  microscope 
the  movable  upper  lenses 


FIG.  313. — Microscope  Model  II  a.    (Fuess.) 


1  Fred  Eugene  Wright:  Neuere  Verbesserungen  am  petrographischen  Mikroskop.  Cen- 
tralbl.  f.  Min.  etc.,  1911,  581-584. 

C.  Leiss:  Ueber  zwei  neue  Mikroskope  ftir  petrographische  und  krystalloptische  Studien. 
Zeitschr.  f.  Kryst.,  XLIX  (1911),  198. 

See  also: 

Fred.  Eugene  Wright:  A  new  petrographic  microscope.  Amer.  Jour.  Sci.,  XXIX  (1910), 
407-414. 

Idem:  Methods  of  petrographic-microscopic  research.  Carnegie  Publication  No.  158. 
Washington,  1911,  10-13. 


208 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  174 


of  the  condensing  system,  used  in  the  other  Fuess  microscopes,  are 
omitted,  the  Abbe  illuminating  apparatus  making  these  unnecessary. 
At  the  upper  end  of  the  tube  is  a  slide,  similar  to  the  Seidentopf  com- 
pensator (Fig.  469,)  for  the  insertion  of  accessories  in  the  focal  plane  of 
the  ocular. 

174.  Zeiss  Crystallographic  and  Petrographic  Microscope,  HI  MD. — The 

Zeiss1    Crystallographic  and  petrographic    microscope  III  MD   (Fig.  314), 

has  coarse  and  fine  ad- 
justments, the  latter  of 
the  Berger2  type,  one  divi- 
sion of  the  milled  head 
corresponding  to  a  varia- 
tion of  0.002  mm.  in  the 
position  of  the  tube.  The 
inner  tube  is  movable  by 
means  of  a  rack  and  pin- 
ion, and  there  are  milli- 
meter divisions  for  re- 
cording the  tube  length. 
The  Bertrand  lens  is  in- 
serted in  the  lower  end 
of  the  tube.  There  is  a 
polarizer  which  swings 
out,  an  iris  diaphragm 
below  the  stage,  and  a 
condensing  system  of  1.40 
numerical  aperture.  Two 
analyzers  are  provided, 
one  to  swing  out,  and  a 
cap.  There  is  no  upper 
diaphragm  nor  are  there 
means  for  simultaneously 
rotating  the  nicols.  The 
non-mechanical  stage  is 
of  the  revolving  type, 
and  is  graduated.  Above 
the  objective  there  is  a 


FIG.  314.— Microscope  model  III  MD.     (Zeiss.) 


carrier  in  which  the  accessories  may  be  inserted. 

1  S.     Czapski:  Mikroskope  von  Carl  Zeiss  in  Jena  fiir  krystallographische  und  petro- 
graphische  Untersuchungen.     Zeitschr.  f.  Instrum.,  XI  (1891),  94-99. 

2  Max   Berger:   Rin  neucr  Mikroskop-Obcrbau,     Zeitschr.  f.  Instrum.,  XVIII  (1898), 
129-133. 


ART.  176] 


VARIOUS  MODERN  MICROSCOPES 


209 


175.  Zeiss  Small  Mineralogical  Stand  VM. — Smaller  than  the  above  is 
the  microscope  shown  in  Fig.  315.     This  instrument  has  a  two-lens  condens- 
ing apparatus  with  a  numerical  opening  of  i.o,  which,  with  the  polarizer,  may 
be  entirely  thrown  out  of  the  axis  of  the  microscope.     The  lower  lens  of  the 
condenser  is  attached  to  the  polarizer,  the  upper  is  loosely  placed  above  it 
and  may  be  lifted  out  when  the  condenser  is  swung  aside.     This  does  not 
allow  a  very  rapid  change  from  par- 
allel to  convergent  light  or  vice  versa, 

since  it  is  necessary  to  rack  down 
the  polarizer,  swing  it  aside,  insert 
the  condensing  lens  writh  the  fingers, 
swing  it  back,  and  rack  it  up.  The 
polarizer  is  held  in  place  by  friction 
and  may  be  rotated  through  360°. 
In  the  lower  part  of  the  tube  there 
are  two  slides,  one  for  the  analyzer, 
the  other  containing  a  circular  open- 
ing into  which  is  laid  the  gypsum  or 
mica  plate.  There  is  neither  draw- 
tube,  Bertrand  lens,  nor  upper  dia- 
phragm. 

176.  Reichert  Mineralogical 
Stand    MI. — The    Reichert    micro- 
scope MI  (Fig.  316)  has  a  wide  tube 
so  that  it  may  be  used  for  photo- 
micrographic  as  well  as  for  ordinary 
petrographic  work.     It  possesses  a 
rotating  upper  nicol  with  degree  divi- 
sions, a  Bertrand  lens  with  iris  dia- 
phragm, a  slot  at  45°  for  the  inser- 
tion of  the  accessories,  and  an  objective  clutch.     The  revolving  stage,  125 
mm.  in  diameter,  may  be  read  to  minutes,  and  may  be  slowly  rotated,  for 
the  exact  measurement  of  small  angles,  by  means  of  a  tangent  screw  which 
may  be  snapped  into  place.     The  mechanical  stage  is  provided  with  ver- 
niers at  right  angles  to  each  other  and  reading  to  o.oi  mm.     The  stand  is 
large  and  has  a  heavy  foot.     The  coarse  adjustment  is  produced  by  means 
of  a  rack  and  pinion,  the  fine  (Fig.  248),  which  is  located  above  the  arm  so 
that  the  instrument  may  be  carried  by  the  latter,  by  means  of  a  horizontal 
disk,  thicker  at  one  side  than  at  the  other,  thus  forcing  upward  a  wheel 
carrying  the  tube.     It  may  be  read  to  o.ooi  mm.     Beneath  the  stage  is  a 
triple  condensing  system,  the  upper  two  lenses  of  which  may  be  readily 

14 


FIG.  315. — Small  mineralogical  stand  VM. 
(Zeiss.) 


210 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  176 


thrown  out  of  the  line  of  collimation, l  thus  changing  the  light  from  con- 
vergent to  parallel.     The  polarizer,  with  its  iris  diaphragm,  may  be  raised 


-•:,%•<-..  -—  -,7  :::      -     - 

FIG.  316. — Miner alogical  stand  MI.      (Reichert.) 


or  lowered  by  means  of  a  milled  head,  and  both  may  be  removed  and  re- 
placed by  an  Abbe  illuminating  apparatus. 

1  The  line  of  collimation  is  the  line  joining  the  intersection  of  the  cross-hairs  and  the 
optical  center  of  the  objective. 


ART.  178] 


VARIOUS  MODERN  MICROSCOPES 


211 


177.  Reichert  Mineralogical  Microscope  MVin. — A  smaller  and  cheaper 
microscope  is  shown  in  Fig.  317.     The  stand  is  intermediate  in  form  be- 
tween the  German  horseshoe  and  the  English  tripod,  and  is  non-tilting.    The 
illuminating  apparatus,  condenser,  and  iris  diaphragm  may  be  swung  entirely 
aside  by  means  of  the  vertical  screw  shown  beneath  the  stage.     The  upper  two 
lenses  of  the  condensing 

lens  may  be  moved  aside 
independently  to  change 
from  convergent  to  par- 
allel light.  Focussing  is 
by  means  of  a  rack  and 
pinion,  there  being  no 
fine  adjustment.  In  the 
tube,  which  is  of  fixed 
length,  are  inserted  the 
analyzer  and  Bertrand 
lens  on  sliders,  so  that 
they  may  be  readily  in- 
serted or  removed.  The 
centering  screws  for  the 
objective  work  at  45°, 
and  there  is  an  objective 
clutch  beneath  them. 

178.  Bausch  &  Lomb 
LCH  Petrographic  Mi- 
croscope.— The  Bausch 
&  Lomb  LCH  stand  (Fig. 
318)  has  recently  been 
improved  and  is  now  ca- 
pable of  doing  most  of 
the  work  ordinarily  re- 
quired of  a  petrographic 
microscope.     The  space 
above  the  stage  is  large, 
giving  ample  room  for 

the  USe  Of  Stage  aCCeSSO-  FlG    3I7._Mineralogical  stand  M  VIII.     (Reichert.) 

ries.  The  friction  draw- 
tube  is  graduated  to  millimeters,  and  has  a  slot  for  the  Bertrand  lens  with  an 
iris  diaphragm  above  it.  The  oculars  are  of  standard  size  (23  mm.  in  diam- 
eter). The  upper  nicol  is  capable  of  being  rotated  90°,  the  lower  nicol  360°. 
There  are  centering  screws  above  the  objective  working  parallel  to  the  cross- 
hairs, an  objective  clutch,  and  a  fine  adjustment  screw  of  the  lever  type  (Fig. 
249)  reading  to  0.0005  mm-  No  strain  comes  on  the  adjustment  screw  when 


212 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  178 


the  instrument  is  lifted,  the  mechanism  being  above  the  handle.  The  stage 
is  90  mm.  inside,  and  102  mm.  outside  the  degree  graduations,  and  the 
rotation  is  read  by  means  of  a  vernier  reading  to  0.1°.  The  polarizer,  a 
nicol  prism  with  an  angular  field  of  19°,  may  be  swung  entirely  out  of  the 


FIG.  318. — Petrographic  microscope  LCH.     (Bausch  and  Lomb.) 

optical  axis  when  desired,  which  is  a  good  point.  The  upper  lenses  of  the 
condenser,  which  is  of  three  lenses  and  N.  A.  i.io,  may  be  thrown  out  of 
the  axis  of  the  microscope  without  disturbing  the  polarizer  or  the  iris  dia- 
phragm. A  mechanical  stage  may  be  substituted  for  the  one  regularly  used. 


ART.  179] 


VARIOUS  MODERN  MICROSCOPES 


213 


179.  Nachet  Microscope. — The  Nachet1  microscope  (Fig.  319)  is  quite 
different,  in  some  respects,  from  any  of  the  instruments  described  above. 
The  objective  is  connected,  by  means  of  a  separate  arm,  with  the  rotating 
stage,  making  re-centering  unnecessary,  when  using  different  objectives, 


FIG.  319. — Petrographic  microscope.     (Xachet.) 

since  the  whole  arm  rotates  with  the  stage.  This  arrangement  is  convenient 
for  centering  small  mineral  fragments,  but  the  arm  attached  to  the  stage 
is  in  the  way  when  one  wishes  to  use  certain  accessory  apparatus,  such  as 

1  A.  Xachet:    On  a  petro graphical  microscope.     Jour.  Roy.  Microsc.  Soc.,  Ill  (1880), 
227-228. 

Anon:  Nachet's  petro  graphical  microscope.     Ibidem,  N.  S.,  I  (1881)  934-935. 


214 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  179 


that  of  von  Fedorow,  etc.  Focussing  is  also  inconvenient,  since  the  milled 
head  is  not  always  in  the  same  place.  The  stage  is  divided  into  degrees  but 
may  be  read  to  6  minutes  by  means  of  verniers.  It  may  be  rotated  by  hand 


PIG.  320. — Improved  Dick  petrographic  microscope.     (Swift  and  Son.) 

or  by  means  of  a  tangent  screw  on  the  opposite  side  of  the  microscope  from 
that  shown  in  the  figure.  The  mechanical  stage  is  moved  by  two  screws 
working  at  right  angles  to  each  other.  The  analyzer  may  be  swung  out  on 
a  hinge,  the  polarizer  on  a  pivot,  as  shown  in  Figs.  256  and  319. 


ART.  1811  VARIOUS  MODERN  MICROSCOPES  215 

1 80.  Swift's  Improved  Dick  Petrographic  Microscope. — As  mentioned 
above,1   the    first   petrographic    microscope   with    simultaneously    rotating 
nicols  was  designed  by  Allan  B.  Dick2  and  was  made  by  James  Swift  &  Son, 
London.     In  considerably  improved  form,3  it  is  still  made  by  the  same  firm 
(Fig.  320).     The  stage  is  fixed  but,  if  desired,  a  rotating  stage  may  be  at- 
tached.    The  polarizer  O,  the  analyzer  A  of  the  cap  variety,  and  the  ocular 
B  with  its  cross-hairs,  may  be  rotated  together  by  means  of  the  wheel  -E,  which 
may  be  clamped  in  any  position  by  means  of  a  small  screw  at  the  back.    The 
amount  of  rotation  may  be  read  to  5  minutes  by  means  of  a  hinged  magnifier 
D.     Either  nicol  may  be  rotated  independently  or  thrown  out  of  the  line 
of  collimation.     An  alternative  analyzer  H  is  fitted  in  the  tube,  as  are  also 
two  Bertrand  lenses  (F,  G),  the  lower  giving  a  large,  the  upper  a  small  inter- 
ference figure.     The  upper  Bertrand  F  is  fitted  with  a  rotating  diaphragm 
with  six  openings  of  different  sizes.     Beneath  the  stage  is  a  triple  revolver 
carrying  three  different  condensers  K  and  an  iris  diaphragm  M.    N  are 
handles  for  rotating  a  disk  into  which  may  be  set  a  variety  of  stops,  and 
the  whole  condensing  system  may  be  raised  or  lowered  by  means  of  the  screw 
P.    The  microscope  has  both  coarse  and  fine  adjustment  (/),  the  latter 
reading  to  a  thousandth  of  a  millimeter  of  vertical  movement.     C  is  a  slot 
for  accessories  and  D  a  lens  for  reading  the  amount  of  rotation  of  the  nicols. 

181.  Swift's    Large    Petrographic    Microscope. — Another    microscope 
manufactured  by  Swift  &  Son  is  shown  in  Fig.  321.     It  has  a  large  tube 
and  was  designed  especially  for  photomicroscopy  but  can  be  used  equally 
well  for  all  purposes.     It  differs  from  the  other  microscopes  here  described 
in  its  hinge,  which  is  so  constructed  that  the  center  of  gravity  remains  low 
down,  however  the  body  may  be  inclined.4     It  may  be  clamped  in  any 
position  by  means  of  the  screw  T,  and  possesses  the  advantage  that  it  is 
impossible  to  overturn  it  backward.    The  mechanical  stage,  whose  rotation 
may  be  read  to  5  minutes  by  means  of  verniers,  may  be  clamped  by  the  screw 
N  in  any  position.     The  upper  end  of  the  large  tube  may  be  removed  by 
unfastening  the  screw  F,  and  a  photographic  lens  inserted,  the  large  tube 
preventing  the  cutting  off  of  the  outside  rays.    The  fine  adjustment  screw 

1  Article  139,  supra. 

2  Allan  Dick:    A  new  form  of  microscope.     Mineralog.  Mag.,  VIII  (1888),  160-163. 
Idem:  Notes  on  a  new  form  of  polarizing  microscope.    London,  1890.* 

Idem:  Additional  notes  on  the  polarizing  microscope.    London,  1894.* 

Anon:  Dick  and  Swift's  patent  petrological  microscope.    Jour.  Roy.  Microsc.  Soc.,  1889, 

432-436. 

Anon:  Messrs.  Swift  and  Son's  improved  Dick  petrological  microscope.     Ibidem,  1895, 

97- 

3  G.  W.  Grabham:    An  improved  form  of  petrological  microscope,  etc.      Mineralog.  Mag. 
XV  (1910),  335-338,  347-348. 

*  (-Wenham):  A  new  microscope.     Northern  Microsc.,  II  (1882),  108-110. 


216 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  181 


reads  to  0.002  mm.  The  polarizer  S  is  large,  has  a  graduated  lower  flange, 
and  may  be  rotated.  There  are  two  analyzers,  both  of  the  Glan-Thompson 
type.  The  lower  one  swings  in  or  out  by  means  of  the  lever  K.  The  upper 
one  A  may  be  revolved,  the  amount  of  rotation  being  indicated  on  a  scale. 


FlG.  321. — Large  petrographic  microscope.      (Swift  and  Son.) 

There  are  also  two  Bertrand  lenses  L  and  E.  P  is  an  iris  diaphragm,  Q 
the  lever  and  OO  two  centering  screws  for  the  condensing  system,  and  R 
a  milled  head  by  means  of  which  this  system  may  be  raised  or  lowered. 


ART.  182] 


VARIOUS  MODERN  MICROSCOPES 


217 


182.  Beck's  London  Petrographic  Microscope. — Beck's  "London" 
petrographical  microscope,  large  model  (Fig.  322),  belongs  to  the  class  of 
instruments  having  nicols  simultaneously  rotating.  The  base  is  large,  and 
the  pillar  A  is  so  placed  that  when  the  instrument  is  inclined,  it  will  not 
overbalance.  The  stage  B  is  square  and  non-rotating.  While  a  revolving 
stage  is  not  a  necessity  in  a  microscope  whose  nicols  rotate  together,  it  is  some- 


K 


FIG.  322. — London  petrographic  microscope.     (Beck.) 

times  a  great  convenience.  The  analyzer  and  polarizer  rotate  by  means  of 
the  geared  wheels  G,  and  may  be  clamped  in  any  position.  The  upper  wheel 
G  is  graduated  to  degrees,  and  indicates  the  position  of  the  nicols.  The 
polarizer  O  may  be  revolved  independently  of  the  analyzer,  if  desired,  and 
may  be  swung  out,  as  shown  by  the  dotted  lines.  The  analyzer  K  is  of  the 
cap  variety,  and  may  be  clapped  back  as  shown.  An  objection  to  the  nicol 
prism  above  the  eyepiece  is  that  it  greatly  cuts  down  the  field  of  view,  a 


218  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  183 

nicol  within  the  body  being  preferable.  Two  forms  of  condensers  may  be 
obtained.  The  simpler  form  consists  of  a  hemispherical  lens,  fitted  in  the 
top  of  the  nicol  sleeve,  and  a  second  hemispherical  lens,  pivoted  on  one  side 
of  the  stage  and  capable  of  being  swung  out  of  the  line  of  collimation  by 
means  of  a  lever.  The  larger  condenser  has  a  pivoted  top  lens  which  may  be 
swung  out  of  line.  It  also  carries  an  iris  diaphragm,  and  the  whole  con- 
densing system  may  be  raised  or  lowered  by  means  of  the  milled  head  T. 
In  the  inner  tube  of  the  microscope  is  a  slot  P  carrying  a  Bertrand  lens  R 
and  an  iris  diaphragm  S.  The  accessories  may  be  inserted  in  the  slot  L, 
above  the  eyepiece,  or  in  the  one  below  it  (N);  both  openings  are  at  45°  with 
the  cross-hairs. 

183.  Socle*  te*  Genevoise  Universal  Microscope. — The  microscope  shown 
in  Figs.  323-325  is  especially  adapted  for  von  Fedorow's  methods,  but  may 
be  transformed  into  an  ordinary  mineralogical  microscope.     In  Fig.  323  it  is 
shown  with  the  objective  clamp  and  fine  adjustment  screw  (C)  attached  to 
the  stage  in  the  manner  of  the  Nachet  microscope,  making  centering  inde- 
pendent of  the  objective.    This  clamp  may  be  removed  by  the  screw  J,  and 
the  objective  inserted  in  the  clamp  p,  making  the  microscope  similar  to  the 
German  instruments  (Fig.  324).     In  this  case  the  objective  is  centered  by 
the  screws  V  (Fig.  325).    As  may  be  seen  from  the  illustrations,  the  stage  is 
of  ample  size,  and  to  it  may  be  clamped  a  von  Fedorow  stage   (Fig.  325). 
In  order  to  overcome  the  necessity  of  raising  the  tube  unduly,  and  thus 
making  the  instrument  top  heavy,  it  is  here  possible  to  lower  the  entire 
stage  by  the  screw  I,  Fig.  323.    This  makes  it  possible,  likewise,  to  use  the 
instrument  as  a  metallographic  microscope. 

Besides  these  special  features,  the  instrument  possesses  most  of  the 
attachments  of  the  microscopes  described  above  except  simultaneously  rotat- 
ing nicols.  When  used  with  the  stage  fine  adjustment,  as  shown  in  Fig.  323, 
the  Bertrand  lens  V  is  inserted  in  the  clutch  p;  when  p  is  used  for  the  objec- 
tive, it  is  inserted  in  A .  The  nicol  prisms  are  both  capable  of  being  rotated. 
At  the  upper  end  of  the  tube  is  a  slot  q  at  45°  to  the  cross-hairs  for  the  inser- 
tion of  the  accessories,  and  corresponding  to  it  there  is  one  in  the  focal  plane 
of  one  of  the  Huygens  oculars.  By  means  of  the  milled  wheel  a  the  tube  may 
be  extended.  Rotation  of  the  stage  may  be  read  from  verniers,  a  tangent 
screw  assisting  in  obtaining  fine  adjustments.  Upper  and  lower  diaphragms 
are  provided. 

184.  Fuess  Microscope  for  the  Theodolite  Method. — A  microscope  of  an 
entirely  different  type1  is  shown  in  Fig.  326.     It  combines  in  itself  a  von 
Fedorow  stage  and  a  petrographic  microscope  with  simultaneously  rotating 
nicols.     The  universal  stage  in  this  instrument,  however,  is  considerably 

1  C.  Leiss:  Neue  petrographisches  Mikroskop  fur  die  Theodolit-Methode.  Centralbl.  f. 
Min.  etc.,  1912,  733~736. 


ART.  184] 


VARIOUS  MODERN  MICROSCOPES 


219 


220 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  184 


larger  than  in  the  detachable  stage,  being  capable  of  taking  sections  28X48 
mm.,  thus  doing  away  with  the  necessity  of  using  circular  sections.  The 
construction  of  the  stage  is  similar  to  that  of  the  ordinary  von  Fedorow  stage, 
and  clearly  appears  from  the  illustration.  The  ordinary  rotatory  movement 
of  the  stage  not  being  present,  the  nicols  are  made  to  rotate  simultaneously 


FIG.  326. — Microscope  with  universal  stage.     (Fuess). 

by  the  rigid  bar  n  in  the  same  manner  as  in  the  microscopes  shown  in  Figs. 
312  and  313.  Its  motion  may  be  read,  by  means  of  a  vernier,  to  5  minutes. 
The  upper  nicol,  which  is  a  Glan-Thompson  prism,  may  be  disconnected 
from  the  bar  n  by  raising  the  tube  until  the  bar  a  passes  over  its  end.  The 
instrument  is  made  with  or  without  a  hmge  for  tilting,  and  without  fine 
adjustment  for  focussing. 


ART.  185] 


VARIOUS  MODERN  MICROSCOPES 


221 


185.  Beck's  Rosenhain  Metallurgical  Microscope. — Still  another  type 
of  microscope  is  necessary  for  metallurgical  work  because  an  artificial  source 
of  light  is  generally  used,  and  it  is  inconvenient  to  change  its  position.  For 
this  reason  the  stage  is  made  to  move  by  means  of  a  rack  and  pinion,  thus 
focussing  the  instrument  from  below  without  disturbing  the  tube. 


FIG.  327. — Rosenhain  metallurgical  microscope.     (Beck.) 

A  microscope  of  this  kind  is  the  Rosenhain  metallurgical  microscope, 
shown  in  Fig.  327.  At  the  side  of  the  body  tube  is  an  opening,  guarded  by 
an  iris  diaphragm,  to  regulate  the  amount  of  light  admitted.  This  instru- 
ment may  be  used  for  examining  polished  faces  of  rocks  as  well  as  of  metals. 


CHAPTER  XII 
SELECTING,  USING,  AND  TAKING  CARE  OF  A  MICROSCOPE 

186.  Selecting  a  Microscope. — The  general  requirements  of  a  good  petro- 
graphic  microscope  are  thus  summarized  by  Wright:1 

"  (i)  Firm,  rigid  stand  for  the  support  of  the  optical  system. 

"  (2)  Optical  system  centered;  optic  axis  of  the  system  to  pass  through  the  center 
of  rotation  of  the  stage. 

"  (3)  Simple  device  for  centering  the  objective;  the  centering  screws  to  be  par- 
allel with,  and  not  diagonal  to,  the  cross-hairs  of  the  ocular  in  order  that  the  observer 
may  have  field  coordinates  as  guides.  To  center  the  stage  instead  of  the  objective 
is  wrong  in  principle  as  it  displaces  the  one  point  to  which  the  optical  system  is  tied. 

"  (4)  Easy  passage  from  parallel  to  convergent  polarized  light. 

"  (5)  Easy  passage  from  low  to  high  powers. 

"  (6)  Bertrand  lens  centered  and  adjusted  to  proper  focus. 

"(7)  Properly  constructed  coarse  and  fine  adjustment  screws  for  focussing  the 
objective,  the  fine  adjustment  screws  to  record  intervals  of  o.ooi  mm.  and  to  be 
free  from  lost  motion. 

"  (8)  Satisfactory  arrangement  for  raising  and  lowering  the  sub-stage  condenser. 

"(9)  Accurately  constructed  mechanical  stage  on  which  lateral  movements  of 
o.oi  mm.  can  be  measured  directly. 

"  (10)  Degree  circle  of  stage  to  be  accurately  divided  and  provided  with  vernier 
to  read  to  5'  at  least. 

"(n)  The  ocular,  the  upper  nicol  carriage,  the  Bertrand  lens  support — in 
short,  all  moving  parts — to  fit  accurately,  so  that  on  insertion  they  invariably 
return  to  exactly  the  same  point." 

To  these  points  may  be  added,  accessible  adjustment  screws,  plainly 
readable  stage  vernier,  lower  diaphragm,  and  diaphragm  above  the  Bertrand 
lens.  It  is  desirable  also  to  have  readily  removable  lower  nicol,  and  nicols 
simultaneously  rotating. 

Which  microscope  to  choose,  from  among  the  numerous  instruments  on 
the  market,  depends  largely  upon  the  use  to  be  made  of  it,  and  upon  the 
amount  of  money  which  is  to  be  spent.  The  microscope  which  is  chosen 
for  individual  use,  and  which  can  have  the  personal  care  of  the  owner,  may 
not  be  the  instrument  one  would  put  in  the  hands  of  a  miscellaneous  lot  of 
undergraduates.  Men  doing  advanced  work  require  more  elaborate  micro- 
scopes, instruments  capable  of  taking  all  of  the  attachments  which  may  aid 

1  Fred  Eugene  Wright:  The  methods  of  petrographic-microscopic  research.  Carnegie 
Publication  No.  158.  Washington,  D.  C.,  1911,  12-13. 

222 


ART.  187]  USE  AND  CARE  OF  A  MICROSCOPE  223 

in  research.  It  may  be  that  a  single  instrument  will  not  answer  the  purpose, 
and  several  microscopes,  adapted  to  specific  uses,  must  be  purchased.  So 
far  as  the  cost  is  concerned,  if  a  certain  selection  of  accessories  will  tempo- 
rarily answer  the  purpose,  a  better  grade  of  instrument  may  be  purchased 
and  additional  equipment  added  as  occasion  demands.  A  study  of  cata- 
logues, and  a  comparison  of  the  instruments  described  in  the  previous  chapter, 
may  help  in  making  a  selection.  The  instrument  to  which  one  is  accustomed 
is  likely  to  appear  the  most  satisfactory. 

For  students'  use  the  following  equipment  is  sufficient  for  most  purposes. 

A  stand  having  coarse  and  fine  adjustment,  revolving  stage,  upper  and 
lower  nicols,  upper  and  lower  diaphragms,  condensing  lens,  attached  Bert- 
rand  lens,  objective  clutch,  centering  device,  and  slot  for  accessories. 

Two  Huygens  oculars  with  magnifying  powers  of  5  and  10  times.  A 
micrometer  ocular  with  a  magnification  of  7.5  is  often  useful. 

Three  objectives  of  approximately  the  following  focal  lengths,  40  mm., 
15  mm.,  and  5  mm.  (See  table,  Art.  153.) 

A  quartz  wedge  and  a  selenite  plate  or  a  combination  wedge  such  as  is 
described  in  Article  298. 

An  objective  of  approximately  2.5  mm.  focal  length  is  often  desirable 
for  obtaining  interference  figures  on  small  particles,  but  it  is  not  at  all  neces- 
sary that  each  student's  outfit  should  be  equipped  with  one.  If  all  of  the 
microscopes  used  in  a  laboratory  are  of  the  same  kind,  it  will  be  found  that  a 
single  example  of  many  accessories,  such  as  special  oculars,  objectives, 
markers,  etc.,  may  be  used  in  common  by  all  the  students. 

USE  AND  CARE  OF  A  MICROSCOPE 

187.  Light. — The  best  light  for  microscopic  work  is  that  coming  from  the 
north;  next  best,  from  the  east.  There  should  be  no  obstructing  buildings 
or  trees,  and  the  mirror  of  the  microscope  should  be  able  to  reflect  direct 
light  from  the  sky.  Direct  sunlight  should  never  be  used. 

On  dark  days,  or  where  it  is  impossible  to  obtain  proper  daylight,  an 
artificial  light  is  a  great  convenience.  The  source  of  the  light  is  immaterial 
provided  that  it  is  strong  enough,  and  that  it  is  properly  corrected  for  color. 
If  not  corrected,  the  interference  colors  will  appear  abnormal  and  the  light 
be  unsatisfactory.  The  usual  artificial  lights  are  all  too  yellow  and  must, 
consequently,  be  corrected  by  a  blue  glass  of  proper  intensity. 

The  lamp  shown  in  Fig.  328,  after  Dr.  O.  Lassar,1  is  made  for  the  use  of 
oil  or  gas.  With  the  latter,  an  Auer  burner  is  used.  It  has  a  silvered  re- 
flector and  a  cobalt  blue-glass  front.  The  light  is  approximately  of  the  tone 
of  daylight  but  is  hardly  strong  enough. 

1  Similar  lamps  are  described:  Parkes's  microscope  lamp  with  cooling  evaporator.  Jour. 
Roy.  Microsc.  Soc.,  Ill  (1880),  528-529;  and  Schieck's  microscope  lamps.  Ibidem.  1888. 
490-491. 


224 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  187 


A  light  made  by  Baker1  consists  of  a  Nernst  electric  lamp  mounted  on  a 
heavy  tripod,  and  capable  of  being  adjusted  to  any  height  or  tilted  to  any 
angle.  The  globe  covering  the  light  is  blackened  except  a  small  aperture 
in  front  through  which  the  light  passes.  Colored  screens  are  used  to  modify 
the  light.  A  similar  lamp,  with  a  6o-watt  incandescent  bulb,  and  provided 
with  blue,  amber,  and  diffusing  screens,  is  manufactured  by  Leitz  (Fig.  329). 

Wright2  described  an  acetylene  gas  burner,  fed  by  a  J.  B.  Colt  generator 
No.  102,  and  the  writer  has  used  both  a  Nernst  and  an  80- Watt  ii5~volt 
tantalum  lamp,  properly  shaded,  toned  down  by  cobalt  glass,  and  made 


FlG.  328. — Microscope  lamp 
after  Dr.  O.  Lassar.  1/7  nat- 
ural size.  (Fuess.) 


FIG.  329. — Artificial  light.     (Leitz.) 


uniform  by  a  finely  ground  glass  screen.  Either  light  is  of  sufficient  strength, 
but  the  latter  is  too  fragile  if  handled  much.  If  attached  to  the  wall  where 
it  is  not  likely  to  be  jarred,  it  makes  an  ideal  light. 

Between  any  artificial  source  of  light  and  the  mirror,  there  should  be 
placed  a  condensing  lens  of  some  sort,  in  order  that  the  beams  may  be  col- 
lected, although,  as  mentioned  above,  a  ground-glass  screen  will  do  fairly 
well.  This  condenser  may  be  nothing  more  than  a  Florence  flask,  15  to  20 
cm.  in  diameter,  and  filled  with  water  or  an  ammonia  copper  sulphate  solu- 
tion, made  by  adding  50  c.c.  of  ammonia  to  25  c.c.  of  a  10  per  cent,  copper 
sulphate  solution,  and  then  diluting  it  to  fill  a  6-in.  flask.3  If  the  solution  is 

1  Anon:  C.  Baker's  electric  lamp  for  the  microscope.     Jour.  Roy.  Microsc.  Soc.,  1905,  252. 

2  Fred.  Eugene  Wright:  Artificial  daylight  Jor  use  with  the  microscope.     Amer.  Jour. 
Sci.,  X  (1909). 

3  Charles  J.  Chamberlain:  An  artificial  light  j 'or  the  microscope.     Jour.  Appl.  Microsc., 
VI  (1903),  2663-5. 


ART.  190] 


USE  AND  CARE  OF  A  MICROSCOPE 


225 


milky,  add  more  ammonia.  For  class  work,  three  or  four  globes  may  be 
used  around  one  open  light,  such  as  a  Welsbach  burner.  The  globes  should 
partly  project  through  circular  openings  in  blackened  screens,  which  thus 
serve  to  keep  out  all  direct  light.1 

More  convenient  than  a  glass  globe,  and  not  expensive,  is  a  bull's-eye 
condenser  (Fig.  330),  75  to  100  mm.  in  diameter.  If  mounted  on  a  stand  as 
shown  in  the  illustration,  it  may  be  adjusted  to  any  height  or  any  angle. 

The  position  of  the  artificial  light  is  a  matter  of  convenience,  and  it 
may  be  placed  either  to  the  front  or  at  one  side. 
With  a  light  which  requires  attention,  it  is  most 
convenient  to  have  it  at  the  side. 

188.  Table.— The  table  should  be  firm  and 
of  a  height  to  suit  the  convenience  of  the  in- 
dividual. If  one  works  with  the  microscope 
inclined,  a  height  of  28  to  30  in.  (71  to  76 
mm.),  and  used  with  a  chair  of  17  to  17  1/2  in. 
(43  to  44  1/2  mm.),  is  generally  satisfactory. 
If  the  instrument  is  used  upright,  the  table 
must  be  lower.  In  the  laboratory,  a  long  table 
attached  to  and  extending  the  length  of  the 
north  wall  will  accommodate  the  greatest  num- 
ber of  students.  It  should,  however,  be  ex- 
tremely rigid  and  firmly  attached,  so  that  no  jar  will  be  transmitted  from 
one  part  to  another.  In  height  it  may  be  36  in.  thereby  permitting  a  student 
to  stand  or  to  regulate  the  height  of  his  revolving  stool  as  he  finds  most  rest- 
ful, and  at  the  same  time  allowing  the  instructor  to  glance  through  the  in- 
strument with  the  least  possible  disturbance  to  a  class.  The  working  table 
should  be  fitted  with  drawers  in  which  to  keep  accessories,  and  a  cabinet 
or  bell  jar  should  be  provided  to  protect  the  microscope  from  the  dust. 
For  laboratory  classes,  it  is  also  desirable  that  at  least  one  artificial  light 
be  provided  for  each  two  students. 


FIG.     330. — Bull's-eye   condenser. 
(Central  Scientific  Co.,  Chicago.) 


METHOD  OF  WORKING 

189.  Position. — The  least  possible  fatigue  will  be  felt  by  the  student  if 
he  sits  perfectly  upright,  with  the  arms  resting  on  the  table,  and  so  places 
the  microscope  that  it  will  not  be  necessary  to  compress  the  chest  or  strain 
the  neck  in  looking  through  it.     The  instrument  should  be  placed  squarely 
in  front,  so  that  both  hands  may  be  used  to  manipulate  it. 

190.  Proper  Eye  to  Use. — Use  whichever  eye  is  least  fatigued  by  the 
work,  and  keep  the  other  eye  open.     It  may  be  difficult,  at  first,  not  to  see 


1  A  condenser  of  the  kind  here  described  is  made  by  Bausch  and  Lomb. 
15 


226  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  191 

with  this  eye,  but  after  a  short  time  no  exertion  will  be  necessary  to  let  it 
remain  passive.  If  both  eyes  can  be  used  equally  well,  make  a  point  of  chang- 
ing from  one  to  the  other.  Keep  the  eye  close  to  the  eyepiece.  The  proper 
position  is  in  the  Ramsden  disk  (EP,  Fig.  229),  which  is  very  close  to  the  eye- 
lens  in  high  powers,  and  slightly  farther  removed  in  low. 

191.  Eye  Shade. — It  not  only  adds  materially  to  the  comfort  of  working 
but  makes  a  brighter  image,  by  allowing  the  pupil  of  the  eye  to  dilate,  if 
much  of  the  outside  light  is  excluded  by  means  of  shades.  If  one  works 
facing  a  window,  a  square  of  black  cloth  hung  over  a  wire,  and  extending 
from  8  to  10  in.  above  the  tube  to  about  the  level  of  the  stage,  is  very 
convenient,  and  may  be  shoved  aside  when  it  is  desired  to  work  by  incident 

light.  Another  good  shade  is  made  by  cutting 
a  3/8-in.  board  into  the  form  shown  at  A,  Fig. 
331.  The  hole  should  fit  the  tube  snugly.  A 
dark  pasteboard  hood  (double-faced  corrugated 
board  does  very  well),  with  a  curtain  reaching 
to  the  stage,  may  be  set  on  this  board  to  exclude 
practically  all  of  the  light.  It  may  be  cut  from 
one  piece  of  paper  as  shown  at  B,  the  heavy 
lines  indicating  the  cuts,  the  dotted  lines,  scor- 
ings along  which  to  bend.  The  narrow  strip  P 
^  P  should  be  bent  upward  and  fastened  at  P' .  For 


FIG.  331.— Eye  shade.  observations  by  incident  light  the  entire  hood, 

but  not  the  board,  should  be  removed.  If  one 

wishes  to  work  with  the  left  eye,  instead  of  the  right,  the  board  A  may  be 
reversed.  If  the  light  comes  from  the  left  side,  instead  of  the  right,  the 
scorings  should  be  made  on  the  other  side  of  the  pasteboard,  and  the  sides 
bent  in  the  opposite  direction.  This  will  bring  the  curtain  P'  on  the  left  side. 

Dr.  J.  Peiser1  described  a  shade  made  as  follows:  A  copper  wire,  2  mm. 
in  thickness  and  25  cm.  long,  is  fastened  to  a  leather-covered  ring  which 
clamps  to  the  tube  below  the  eyepiece.  This  wire  is  curved  backward  and 
upward,  and  at  its  upper  end,  a  hollow  brass  tube,  2  mm.  in  diameter  and 
66  cm.  long  and  bent  into  the  form  of  a  parabola,  is  attached  at  its  center. 
A  black  satin  curtain,  slit  at  the  lower  end  to  form  two  pendants,  which  may 
be  held  up  with  two  snap  fasteners,  is  attached  to  the  cross- wire  and  forms 
the  shade.  A  similar  shade  was  described  previously  by  Schiefferdecker.2 

A  very  simple  shade  is  shown  in  Fig.  33  2. 3    It  is  attached  to  the  upper 

1  J.  Peiser:  Em  Mikroskopierschirm.     Zeitschr.  f.  wiss.  Mikrosk.,  XXI  (1904),  467- 

469. 

2  P.  Schiefferdecker:  Ueber  einen  Mikroskopirschirm.     Ibidem.  IX  (1892),  180-181. 

3  R.  H.  Ward:  An  eye-shade  for  monocular  microscopes.     Amer.  Mon.  Microsc.  Jour., 
V  (1884),  82-83. 

A  similar  shade  was  described  by  E.  Pennock:  Eye  shade  for  monoculars.  Jour.  Roy. 
Microsc.  Soc.,  N.  S.,  I  (1881),  518. 


ART.  195]  USE  AND  CARE  OF  A  MICROSCOPE  227 

part  of  the  tube  of  the  microscope  and  may  be  used  for  either  eye.     Oculars 
may  be  changed  without  removing  it. 

192.  Amount  of  Light. — Use  the  lower  diaphragm  to  cut  off  superfluous 
light,  the  amount  depending  upon  the  objective.     Enough  should  be  ad- 
mitted so  that  structures  may  be  seen  without  straining  the  eye,  but  not 
enough  to  produce  a  glare.     If  too  much  light  is  admitted,  it  conceals  the 
finer  detail.    More  light  should  be  admitted  when  the  nicols  are  inserted 
than  when  they  are  out. 

193.  Proper  Magnifying  Power  to  Use. — Begin  work  with  low-power 
objectives,  and  increase  the  magnification  as  necessary.     Do  as  much  work 
as  possible  with  the  low  powers  and  save  your  eyes.     For  the  greater  part 
of  the  work,  no  magnification  greater  than  50  to  60  diameters  is  necessary. 
For  interference  figures,  180  diameters  is  generally  ample. 

194.  Objective  Clutch. — When  using  an  objective  clutch  of  a  pattern 
similar  to  that  shown  in  Fig.  239,  it  is  advisable  to  give  the  objective  a 
slight  rotation  after  insertion  in  order 

to  insure  its  dropping  into  proper  posi- 
tion. If  the  objective  still  appears 
markedly  out  of  center,  do  not  at  once 
adjust  the  cross-hairs,  but  remove  the 
objective  and  examine  it  and  the  clutch 
for  foreign  matter.  It  is  a  good  plan 
to  wipe  each  objective  collar  when  be-  Fic'  332'~^h  LTSST' 


ginning  the  day's  work. 

195.  Focussing. — Become  familiar  with  the  free  working  distance  of  the 
objectives,  so  that  they  may  automatically  be  set  roughly  in  focus.  For 
high  powers,  place  the  eye  on  a  level  writh  the  stage,  and  look  toward  a 
window  between  the  cover-glass  and  the  lens,  lowering  the  objective  until 
but  a  narrow  streak  of  light  is  seen.  Now  look  through  the  ocular,  and 
raise  the  tube  very  slowly  until  the  section  is  in  focus.  Always  focus  upward 
and  no  thin  sections  will  ever  be  broken.  For  colorless  minerals,  such  as 
quartz,  cut  down  the  illumination,  and  look'  for  bubbles  or  other  inclusions. 
Use  a  low-power  objective  as  a  finder  and  place  the  mineral  desired  under 
the  cross-hairs.  In  removing  high-power  objectives  always  raise  the  tube 
lest  the  cover-glass  or  the  objective  be  injured. 

Various  devices  for  safe-guarding  the  slide  against  breakage  by  the  ocular 
have  been  devised.1  Most  of  them  consist  of  a  ring  about  the  objective,  to 

1  E.  H.  Griffith:  On  several  new  microscopical  accessories.  Proc.  Amer.  Microsc.  Soc., 
9th  meeting,  VIII  (1886),  150-152. 

S.  Gelblum:  Discussion  des  conditions  generates  que  doit  remplir  le  dispositif  d'arret  du 
tube  a,  tirage  dans  tout  microscope,  et  description  du  moycn  pratique  pour  arriver  a  cc  result. 
Zeitschr.  f.  wiss.  Mikrosk.,  XX  (1903),  129-132. 

S.  E.  Dowdy:  A  focussing  safeguard.     English  Mechanic,  LXXVIII  (1903),  291. 


228  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  196 

which  is  attached  a  button  or  pin  which,  upon  lowering  the  tube,  comes  in 
contact  with  the  edge  of  the  slide  or  a  button  on  the  stage. 

196.  Changing  the  Ocular. — When  changing  from  one  ocular  to  another, 
especially  if  they  fit  snugly,  raise  the  tube,  and  take  care  not  to  press  the 
objective  through  the  thin  section.     Not  only  will  the  slide  be  broken  but 
the  objective  may  be  ruined  as  well. 

The'method  of  centering  the  objective  was  described  above  (Art.  114). 

HINTS  ON  THE  CARE  OF  A  MICROSCOPE 

197.  Care  of  the  Stand. — Keep  the  stand  of  the  microscope,  especially 
the  working  parts,  free  from  dust. 

Do  not  carry  the  microscope  by  any  part  above  the  fine  adjustment, 
unless  you  wish  to  ruin  it.  Do  not,  for  example,  carry  microscopes  with 
the  prism  type  of  fine  adjustment  by  the  arm  (Figs.  307,  308,  310,  311,  322). 
They  should  be  carried  by  the  post.  If  the  hinge  only  is  below  the  arm,  the 
latter  is  the  most  convenient  part  by  which  to  carry  it  (Figs.  230,  309,  314, 
316,  318,  etc.). 

Do  not  clean  the  stand  with  alcohol,  for  it  will  remove  the  yellow  lacquer. 
Use  benzene  or  xylene,  and  wipe  with  a  soft  cloth  in  the  direction  of  the 
grain  of  the  metal,  never  across.  If  the  microscope  has  a  vulcanite  stage  and 
the  benzene  stains  it,  clean  it  by  rubbing  with  oil. 

Lubricate  the  working  parts  with  clock  oil.  If  it  becomes  gummy,  clean 
with  benzene  applied  with  a  cloth.  If  the  microscope  has  an  inner  tube, 
occasionally  remove  it,  wet  a  cloth  with  a  small  amount  of  oil,  and  wipe  the 
inside  of  the  outer  and  the  outside  of  the  inner  tube.  Oil  the  slides,  but  not 
the  teeth,  of  the  rack  and  pinion.  The  latter  should  be  kept  free  from  dust 
and  be  cleaned  with  benzene. 

If  any  part  of  the  microscope  is  unscrewed,  use  great  care,  when  replac- 
ing screws,  to  start  the  threads  properly.  If  once  cross-threaded,  the  screw 
is  ruined. 

Use  a  screw-driver  which  is  neither  too  large  nor  too  small,  and  see  that 
it  is  of  the  same  shape  as  the  slot  in  the  screw  head. 

198.  Care  of  Nicols  and  Lenses. — Do  not  expose  a  microscope  to  sudden 
changes  of  temperature.     If  moved  from  a  cold  to  a  warm  room,  moisture  is 
likely  to  gather  on  the  lenses,  or  the  balsam  may  crack. 

Do  not  expose  the  lenses  or  the  nicol  prisms  to  direct  sunlight,  nor  keep 
them  near  a  steam  radiator.  The  cement  may  soften. 

Remember  that  the  nicol  prisms  are  made  of  calcite  which  is  very  soft 
and  likely  to  become  scratched.  Dust  them  only  with  a  soft  camel-hair 
brush.  Remember,  too,  that  they  are  expensive. 

Be  sure  that  the  lens  surfaces  are  clean  and  free  from  dust.     Remove 


ART.  199]  i  SE  AND  CARE  OF  A  MICROSCOPE  229 

dust  particles  from  oculars  and  objectives  with  a  soft  brush  or  by  blowing 
upon  them,  then  wipe,  with  a  circular  motion,  with  a  soft  cloth.  Use  soft 
linen,  never  silk  or  cotton,  and  keep,  in  a  dust-proof  box,  separate  cloths  for 
lenses  and  stand. 

If  finger  marks  or  dust  cannot  be  removed  with  a  dry  cloth,  breathe  upon 
the  lens  and  wipe,  or  wripe  with  a  cloth  moistened  very  slightly  with  benzene 
or  xylene.  Use  great  care  to  prevent  any  of  the  cleaning  fluid  from  getting 
between  the  lenses.  Never  use  alcohol. 

Both  sides  of  the  field-  and  eye-lenses  of  the  ocular  may  be  cleaned  if 
necessary,  but  remember  that  when  the  lenses  are  removed  there  is  nothing 
remaining  to  protect  the  cobwebs. 

Front  and  back  surfaces  of  objectives  may  readily  be  cleaned.  Dust  is 
especially  likely  to  settle  on  the  back  lens.  Internal  surfaces  should  be 
examined  with  a  hand  lens  and,  if  any  cloudiness  exists,  the  objective  may  be 
unscrewed  with  great  care.  It  is  better,  however,  to  return  such  lenses 
to  the  maker.1  If  separated  by  the  owner  they  are  likely  to  become 
decentered,  or  more  dust  may  enter  than  is  removed. 

Objectives  used  with  immersion  oil  should  be  cleaned  immediately 
afterward. 

Do  not  let  the  front  lens  of  an  objective  come  in  violent  contact  with  a 
cover-glass,  and  never  let  an  objective  fall.  In  order  to  permit  the  entrance 
of  as  much  light  as  possible  (Figs.  296-297),  the  amount  of  metal  clasping 
the  edge  is  very  little,  in  some  objectives  none  projects  over  the  rim  of  the 
lower  face,  and  it  is  held  in  place  only  by  the  pressure  at  the  sides,  and 
even  here  by  only  a  very  small  piece. 

TESTING  AND  ADJUSTING  THE  MICROSCOPE  AND  THE  ACCESSORIES 

199.  Cross-hairs. — Some  of  the  explanations  given  in  this  and  the  follow- 
ing sections  may  be  in  advance  of  students  who  have  had  no  preliminary 
work  in  petrography.  The  methods  of  testing  and  adjusting,  which  do  not 
more  properly  belong  elsewhere,  are  inserted  here,  however,  in  order  that  all 
such  methods  may  be  brought  together  under  one  heading  for  easy  reference. 

Cross-hairs:  focussing.     See  Art.  163. 

Cross-hairs:  replacing.     See  Art.  164. 

Cross-hairs:  centering.     See  Art.  114. 

Cross-hairs.  To  set  at  right  angles  to  each  other  and  parallel  to  the  directions 
of  vibration  of  the  nicols. — To  determine  whether  the  cross-hairs  of  the  ocular 
are  set  at  right  angles  to  each  other,  a  mineral  with  straight  cleavage,  an 
object  micrometer,  an  object-slip  with  a  straight  scratch  across  it,  or  some 
such  object  is  placed  upon  the  stage,  and  it  is  rotated  until  one  of  the  cross- 

1  William  Wales:  The  proper  care  and  use  oj  microscope  lenses.  Jour.  N.  Y.  Microsc. 
Soc.,  I  (1885),  113-116. 


230  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  200 

hairs  is  parallel  to  it.  The  stage  vernier  is  now  read,  and  the  stage  rotated 
through  90°.  In  its  new  position  the  line  should  be  parallel  to  the  other 
cross-hair.  The  test  should  be  repeated  a  number  of  times. 

The  cross-hairs  should  not  only  be  at  right  angles  to  each  other  but  parallel 
to  the  principal  sections  of  the  nicols  as  well.  The  nicols  are  first  tested  by 
the  method  given  in  Article  202,  after  which  a  slide  consisting  of  a  mineral 
having  parallel  extinction,  such  as  anhydrite,  anthophyllite,  or  needle-like 
quartz  prisms,1  is  placed  on  the  stage  and  rotated  to  the  position  of  darkness. 
This  position  may  be  observed  by  the  use  of  a  gypsum  test  plate  giving  the 
sensitive  violet.  In  this  position  the  cross-hairs  should  be  parallel  to  the 
cleavage  of  the  mineral.  Repeat  the  operation  ten  or  a  dozen  times  and,  if 
the  cross-hairs  and  nicols  are  not  parallel,  rotate  the  cross-hair  support  by 
means  of  a  spanner. 

To  avoid  the  polarizing  effect  of  the  objective,  it  is  better  to  remove  it 
and  use  only  the  Bertrand  lens  in  combination  with  the  ocular,  thus  leaving 
no  lens  between  the  nicols.  The  test  object  should  be  rather  large  in  this 
case,  since  the  magnification  of  the  ocular  and  Bertrand  lens  is  not  great. 
By  pointing  the  microscope  at  the  sun,  the  point  of  extinction  may  be  seen 
much  more  clearly. 

A  Bertrand  ocular  may  be  used  instead  of  a  unit  retardation  plate  to 
determine  when  the  mineral  is  in  the  position  of  extinction. 

200.  Bertrand  Ocular.     Testing  the  position  of  the  division  lines,  which 
should  be  parallel  to  the  vibration  planes  of  the  nicols. — To  set  the  separating 
lines  of  the  Bertrand  ocular  parallel  to  the  principal  sections  of  the  nicols, 
use  is  made  of  an  anhydrite  or  anthophyllite  section.     The  nicols  are  first 
tested  for  accurate  position  of  crossing  by  some  other  means  than  by  the 
Bertrand  ocular,  after  which  the  mineral  is  placed  on  the  stage  in  the  position 
of  extinction.     Upon  inserting  the  Bertrand  ocular  there  should  be  uniform 
color  in  the  four  quadrants.     If  this  is  not  found,  the  vibration  planes  of  the 
nicols  do  not  coincide  with  the  divisions  of  the  Bertrand  ocular. 

201.  Bertrand  Lens.     Centering. — The  center  of  the  Bertrand  lens  should 
lie  exactly  on  the  axis  of  the  microscope.  If  it  does  so,  the  center  of  the  inter- 
ference cross  of  a  section  of  calcite,  cut  exactly  at  right  angles  to  the  c  axis, 
will  lie  at  the  intersection  of  the  cross-hairs  of  the  ocular.     If  it  does  not  do 
so  it  may  be  corrected  by  means  of  the  centering  screws  inserted  in  the  lens 
mounting.     Be  sure  that  the  test  plate  of  calcite  is  accurately  cut  at  right 
angles  to  the  axis. 

202.  Nicol   Prisms. — Determining   the   vibration   directions   of  the   nicol 
prisms.     See  Art.  140. 

1  E.  Weinschenk:  Eine  M ethode  zur  gcnaue  Justirung  der  Nicol' sche  Prismen.  Zeitschr. 
f.  Kryst.,  XXIV  (1904-5),  581-583. 


ART.  203]  USE  AND  CARE  OF  A  MICROSCOPE  231 

To  set  the  nicol  prisms  at  right  angles  to  each  other. — The  principal 
sections  of  the  nicol  prisms  should  be  perpendicular  to  each  other  as 
well  as  parallel  to  the  cross-hairs.  To  test  this,  use  may  be  made  of 
the  Bertrand  ocular.  The  polarizer  is  inserted  with  its  knife  edge  engaged 
in  the  V  notch  of  the  casing.  The  analyzer  is  shoved  out  of  the  axis  of  the 
microscope  and  a  cap  nicol  is  placed  over  the  Bertrand  ocular  and  set  at  o° 
(or  90°,  depending  upon  the  orientation  of  the  polarizer  and  whether  the  eye 
is  most  sensitive  to  blue  or  orange  tones).  In  this  position  the  four  quadrants 
of  the  ocular  should  appear  exactly  the  same  shade  of  color.  If  they  do 
not  do  so,  and  the  amount  of  rotation  necessary  to  produce  uniform  color  is 
greater  than  1/2°  to  i°,  the  nicol  should  be  rotated  in  its  casing  by  means 
of  a  spanner  or  by  the  set  screws,  if  such  are  provided.  The  most  con- 
venient spanner  for  this  purpose  is  a  cylinder,  at  the  upper  end  of  which  are 
two  projecting  points  which  engage  in  the  notches  in  the  nicol  casing.  The 
spanner  may  be  placed  in  position  and,  since  it  is  in  the  form  of  a  tube,  the 
nicol  may  be  rotated  with  it  while  looking  through  the  microscope.  Great 
care  must  be  observed  not  to  scratch  the  lower  surface  of  the  nicol  when  the 
protecting  glass  is  removed  from  below. 

To  correct  the  analyzer,  the  polarizer  must  be  removed,  the  cap  nicol 
turned  to  the  90°  (or  o°)  position,  and  the  same  process  repeated  as  for  the 
polarizer. 

Another  method  of  setting  polarizer  and  analyzer  at  right  angles,  is  to 
place  upon  the  stage  of  the  microscope  a  cleavage  piece  of  anhydrite  or  an- 
thophyllite,  or  a  prismatic  needle  of  quartz1  mounted  in  balsam.  The 
crystal  is  placed  exactly  parallel  to  one  of  the  cross-hairs.  It  should  appear 
perfectly  dark  between  crossed  nicols.  Now,  leaving  polarizer  and  tube- 
analyzer  in  position,  place  a  cap  nicol  above  the  eyepiece  and  rotate  it.  If 
in  any  position  color  appears  in  the  crystal,  it  indicates  that  the  nicols  are  not 
exactly  crossed  and  should  be  corrected. 

To  test  the  two  analyzers  one  proceeds  in  the  reverse  way,  rotating  the 
polarizer. 

Another  method  is  to  remove  from  the  microscope  the  ocular  and  objective, 
and  unscrew  from  the  top  of  the  polarizer  the  condensing  lens.  If  the 
microscope,  with  nicols  crossed,  is  now  pointed  directly  at  the  sun,  the  posi- 
tion of  maximum  darkness  may  be  determined  within  a  quarter  of  a  degree. 
The  sun  will  appear  as  a  dull  disk  in  the  dark  field. 

203.  Accessories. — Determination  of  the  direction  of  c  in  the  one-fourth 
undulation  mica  plate.  Examine  the  interference  figure  produced  by  the 
mica  plate,  using  it  as  a  mineral  section.  The  axis  of  least  ease  of  vibration 
c  is  the  line  joining  the  loci  of  the  hyperbolae,  b  is  at  right  angles  to  this 
line. 

*E.  Weinschenk:  Op.  cit. 


232  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  203 

Determination  of  the  c  direction  in  the  gypsum-plate  (unit  retardation  plate). 
Examine  the  interference  figure,  using  the  gypsum  plate  as  a  mineral  section. 
The  line  joining  the  quadrants  showing  the  lowest  color  (yellow)  is  the  c 
direction. 

Determination  of  the  c  direction  in  a  quartz  or  mica  wedge.  Use  the  wedge 
as  a  mineral  section  and,  with  a  mica  plate  whose  c  direction  is  known  as  an 
accessory,  determine  the  elongation. 


CHAPTER  XIII 
OBSERVATIONS  BY  ORDINARY  LIGHT 

204.  Ordinary  Light. — When  we  speak  of  ordinary  light,  we  mean  light 
which   has   not   been   polarized,  consequently  to  obtain  such,  both  nicol 
prisms  should  be  removed  from  the  microscope.     As  a  matter  of  fact,  in 
many  instruments  the  lower  nicol  is  removed  with  difficulty,  and  one  makes 
his  observations  by  plane  polarized  light.     For  most  minerals  this  is  of  no 
great  consequence  since  there  is  usually  very  little  difference  in  their  appear- 
ance by  ordinary  or  by  plane  polarized  light.     There  are  certain  minerals, 
however,  as  we  shall  see  later,  whose  colors  differ  with  the  direction  of  light 
vibration,  and  their  true  colors  must  be  determined  by  ordinary  light.     The 
intensity  of  the  unpolarized  light  is  nearly  twice  as  great  as  the  plane  polar- 
ized.   This  occasionally  may  make  it  more  advantageous  to  use  the  former. 

Substances  which  are  to  be  examined  by  ordinary  light  are  of  two  classes, 
transparent  and  opaque. 

Transparent  minerals  are  examined  by  transmitted  light  for  crystal  form, 
cleavage,  and  color.  By  it,  also,  angles,  refractive  indices,  lengths,  and 
thicknesses  are  measured. 

Opaque  minerals  are  examined  by  incident  light  for  crystal  form,  color, 
lustre,  etc. 

205.  Determination  of  Crystal  Form. — Crystal  form,  of  both  transparent 
and  opaque  minerals,  is  determined  in  the  same  way  that  it  would  be  in 
cross-sections  of  large  specimens,  but  while  this  determination  is  of  great 
importance  megascopically,  it  is  of  comparative  unimportance  in  sections  of 
rocks.     In  the  latter,  in  the  majority  of  cases,  individual  crystals  have  not 
had  a  chance  for  undisturbed  development,  but  have  had  their  growth 
hampered  in  all  directions  by  the  growth  of  other  crystals.     In  certain  classes 
of  rocks,  namely  the  porphyries,  the  development  of  certain  individuals  has 
been  more  or  less  perfect,  and  a  study  of  then*  forms  may  sometimes  be  of 
assistance  in  their  determination. 

In  hand  specimens  one  has  to  deal  with  more  or  less  perfect  polyhedrons 
or,  if  cleavage  flakes,  polygons  cut  in  a  few  definite  directions  from  the  solid 
forms.  In  rock  sections  one  has  only  random  cross-sections  from  more' or 
less  distorted  solids  from  which  to  make  determinations,  cross-sections  which 
depend  not  only  upon  the  crystal  form,  but  upon  the  direction  in  which  they 
were  cut,  as  well.  Another  difficulty  is  the  fact  that  sections  of  the  same 
shape  may  be  cut  from  totally  different  crystals.  In  spite  of  these  difficul- 

233 


234 


MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  205 


FIG.  333. — Isometric  system.     Cube  and  sections  cut  from  it. 


FlG.  334-— Isometric  system.      Octahedron  and  sections  cut  from  it. 


H  V 


FIG.  335. — Isometric  system.     Icositetrahedron  and  sections  cut  from  it. 


FIG.  336. — Isometric  system.     Tetrahedron  and  sections  cut  from  it. 


FIG.  337. — Tetragonal  system.     Prism  and  sections  cut  from  it. 


FIG.  338.— Tetragonal  system.     Bipyramid  and  sections  cut  from  it. 


ART.  206] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


235 


ties,  however,  it  is  usually  possible,  by  comparing  a  number  of  sections  in 
the  same  rock  slice,  to  determine  the  form  of  the  crystal  from  which  they 
were  cut. 

A  comparison  of  Figs.  333-341  may  be  of  assistance,  especially  to  those 
who  have  not  made  a  study  of  descriptive  geometry.     It  is  impossible  to 


FIG.  339. — Tetragonal  system.     Bipyramid  and  prism,  and  sections  cut  from  it. 

give  all  sections  which  may  be  cut  from  crystals  of  the  different  systems, 
and  only  a  few  of  the  more  common  forms  are  here  shown.  The  student 
may  work  out  others  for  himself. 

From  these  diagrams  it  may  clearly  be  seen  how  it  is  possible  to  cut  a 
hexagonal  section  from  an  isometric  crystal,  a  square  section  from  one  that 


FIG.  340. — Hexagonal  system.     Bipyramid  and 
sections  cut  from  it. 


PIG.  34  I. — H  exagonal   system. 
Prism  and  section  cut  from  it. 


is  hexagonal,  or  a  triangular  section  from  one  of  any  system.  Too  much 
dependence  must  not  be  placed  on  cross-sections,  therefore,  or  it  may  lead 
to  a  wrong  conclusion  in  regard  to  the  crystal  system  to  which  the  mineral 
belongs. 

206.  Cleavage  and  Parting. — Another  property  of  minerals  which  is  to 
be  observed  by  ordinary  light  is  cleavage,  which  is  developed  in  charac- 
teristic directions  in  a  thin  section  by  the  process  of  grinding.  The  direction 
and  perfection  of  the  cleavage  cracks  depend  upon  the  crystal  system  and 
the  substance  itself.  If  the  mineral  possesses  no  cleavage,  the  cracks 
shown  are  irregular;  if  present,  the  cleavage  lines  are  directions  of  least 
cohesion,  and  the  cracks  follow  these  directions  and  appear  as  parallel  lines 
representing  the  traces  of  the  cleavage  planes. 

Cleavage  is  described  as  perfect  when  the  cracks  are  sharp  and  extend 
uninterruptedly  for  considerable  distances.  This  cleavage  is  found  in  mica, 
fluorite,  etc. 


236  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  206 

With  good,  or  distinct,  cleavage  the  cracks  do  not  continue  uninterrupt- 
edly for  such  great  distances,  but  show  off-sets,  and  then  continue  in 
the  same  direction  as  before.  The  off-sets  may  be  irregular  breaks  but 
more  likely  pass  along  other  cleavage  planes,  as  in  hornblende,  augite,  or 
orthoclase. 

Indistinct,  poor,  or  imperfect  cleavage  is  very  irregular.  While  the  lines 
roughly  follow  certain  directions,  the  cracks  are  more  or  less  uneven.  This 
cleavage  is  well  shown  in  olivine. 

Pinacoidal  cleavage  is  generally  well  developed  in  one  direction  only. 
It  is  well  shown  in  mica.  Prismatic  cleavage  is  usually  parallel  to  two 
planes,  as  in  hornblende  or  augite.  In  certain  minerals  of  the  isometric  and 
hexagonal  systems,  three  good  cleavages  are  developed.  In  the  former 
they  are  at  right  angles  to  each  other,  as  in  galena;  in  the  latter  they  form 
rhombohedrons,  as  in  calcite.  With  either  of  these  cleavages,  however, 
generally  only  two  sets  of  lines  are  shown  in  the  thin  sections,  although  three 
may  be.  Certain  isometric  crystals  have  perfect  octahedral  cleavage, 
fluorite,  for  example. 


FIG.  342. — Apparatus    for    obtaining    cleavage   flakes  of  minerals,  after  Wiilfing.      1/3  natural  size. 

(Fuess.) 

While  cleavage  angles  are  important  in-  the  determination  of  minerals, 
they  must  be  used  with  caution  under  the  microscope,  since  the  angles  de- 
pend upon  the  orientation  of  the  random  section  shown  in  the  rock  slice. 
Where  the  sections  are  cut  at  right  angles  to  the  cleavage  planes,-  the  angles 
are  characteristic.  These  sections  may  be  recognized  by  noting,  on  raising 
or  lowering  the  tube  of  the  microscope,  that  there  is  no  displacement  of  the 
cleavage  cracks.  As  an  example  of  two  totally  different  cleavages  appear- 
ing alike,  amphibole  and  pyroxene  may  be  cited.  In  the  former  the  cleavage 
angle  in  a  section  at  right  angles  to  the  prismatic  faces  is  about  124°,  in  the 
latter  about  93°,  yet  a  section  inclined  about  56°  to  the  normal  will  give,  in 
pyroxene,  an  angle  of  124°.  The  cleavage  cracks,  however,  will  not  be 
perpendicular  to  the  section,  and  will  be  laterally  displaced  upon  changing 
the  focus  from  the  top  to  the  bottom  of  the  slide. 

If,  instead  of  using  random  sections  in  a  rock  slice,  one  employs  cleavage 


ART.  207]  OBSERVATIONS  BY  ORDINARY  LIGHT  237 

fragments,  the  determination  is  much  simplified,  since  the  flat  faces  will  here 
bear  definite  relations  to  the  crystallographic  axes.  In  preparing  such  mineral 
fragments,  one  should  crush,  not  pulverize,  the  mineral.  A  diamond  mortar 
is  convenient.  Chisels  and  an  iron  plate  with  a  guard  ring,  such  as  are 
described  by  Wiilfing,1  may  be  used  for  larger  flakes  (Fig.  342). 

In  some  minerals  there  is  occasionally  developed  a  fracture  parallel  to  a 
certain  direction,  but  the  mineral  cannot  everywhere  be  cleaved  parallel  to 
this  plane.  This  parting,  as  it  is  called,  occurs  along  lines  of  weakness,  such 
as  result  from  shearing,  or  develop  along  gliding  planes.  It  is  usually  well 
shown  in  the  small  apatite  crystals  of  granitic  rocks. 

DETERMINATION  or  REFRACTIVE  INDICES 

207.  Relief. — It  has  already  been  pointed  out  that  there  is  a  constant 
ratio  between  the  angle  of  incidence  and  the  angle  of  refraction  of  light  pass- 
ing from  one  transparent  medium  to  another,  and  that  this  constant,  ex- 
pressed by  the  equation  n=—  —  ,  is  called  the  index  of  refraction.  Under 

the  microscope,  minerals  of  different  indices,  embedded  in  Canada  balsam, 
appear  more  or  less  rough.  These  rough  minerals,  from  their  resemblance 
to  shagreen,  are  said  to  have  shagreen  surfaces,2  an  effect  which  may  be  due, 
hi  part,  to  inequalities  of  the  surface,  each  little  elevation  and  depression 
reflecting  and  refracting  the  light  at  a  different  angle,  with  the  result  that 
certain  spots  are  more,  and  others 

less,    illuminated.     It    follows    from     ^-^       i       ^-\       i       /\       \       / 
the  indices  of  refraction  and  critical 
angles  of  two  media,  that  the  greater 
the    difference    between    them,    the 

FIG.  343. — Relief  in  minerals, 

greater   the   contrast  of  the  surface 

inequalities  and  the  rougher  it  appears,  whether  the  mineral  be  of  a  consider- 
ably higher  or  of  a  considerably  lower  index  than  the  balsam.  Another  re- 
sult of  the  difference  in  indices  is  the  apparent  elevation  or  depression  of 
certain  minerals  from  the  plane  of  the  section ;  that  is,  certain  minerals  stand 
out  in  relief.  This  is  due  to  the  fact  that  rays  of  light,  from  the  lower  sur- 
faces of  different  minerals,  appear  to  come  from  the  points  of  intersection 
of  the  refracted  rays  (Fig.  343),  consequently  the  minerals  which  have  a 
higher  refractive  index  appear  to  stand  out  above  the  others. 

If  there  be  placed  upon  a  thin  section  of  a  colorless  mineral  with  a  rough 
surface,  and  without  a  cover-glass,  a  drop  of  a  liquid  with  an  index  of  refrac- 
tion exactly  equal  to  that  of  the  mineral,  it  will  be  found  that  the  appearance 

1  Rosenbusch- Wiilfing:  Mikroskopische  Physiographic,  Ij,  4  Aufl.,  1904,  29. 

2  J.  Thoulet:  De  Vapparence  dite  chagrinee  presentee  par  tin  certain  nombre  de  miner aux 
examines  en  lames  minces.     Bull.  Soc.  Min.  France,  III  (1880),  62-68. 


238  .     MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  208 

of  roughness  disappears,  as  is  to  be  expected,  since  there  will  be  neither  re- 
flection nor  refraction  at  the  contact,  and  the  light  will  pass  through  without 
deflection.  If  a  liquid  with  an  index  either  greater  or  less  be  used,  the  relief 
reappears. 

The  index  of  refraction  of  a  mineral  is  one  of  the  most  important  prop- 
erties for  its  identification,  and  many  methods  have  been  devised  for  its 
determination.  Here  only  those  methods  which  are  applicable  for  use  with 
the  microscope  will  be  discussed. 

There  are  three  microscopic  methods  open  to  the  investigator.  One 
may  determine  the  index  of  refraction  of  the  mineral  directly,  as  by  the  method 
of  the  Due  de  Chaulnes  or  one  of  its  modifications,  one  may  immerse  frag- 
ments of  the  mineral  in  a  fluid  of  known  index,  or  one  may  determine  the 
relation  which  the  refractive  index  of  the  unknown  mineral  bears  to  that  of 
one  which  is  known  and  which  is  in  contact  with  it. 

208.  The  Method  of  the  Due  de  Chaulnes.— The  method  of  the  Due  de 
Chaulnes1  is  one  which  is  applicable  to  the  measurement  of  the  mean  indices 
of  refraction  of  plane-parallel  mineral  plates.  It  depends  upon  the  fact  that 

if  a  medium  or  high-power  objective  is  accu- 
rately focussed  upon  an  object,  and  there  is  in- 
serted between  it  and  the  objective  a  trans- 
parent plate  with  parallel  sides,  the  image  be- 
comes blurred,  and  it  is  necessary  to  raise  the 
tube  of  the  microscope  a  certain  amount  in  order 
that  the  image  may  again  appear  sharp.  The 
FIG.  344.— The  Due  de  chaui-  amount  of  change  necessary  depends  upon  the 

nes'  method  for  measuring  refrac-      index     Qf      refraction    of    the   plate    and    UDOn    its 

live  indices. 

thickness. 

Let  c  d  e  /,  Fig.  344,  be  a  plate  of  an  isotropic  substance  whose  thickness 
has  been  accurately  measured.  A  ray  of  light  Oc  will  be  refracted,  upon 
reaching  the  air  c,  to  the  point  a',  consequently  a  mark  on  the  lower  surface 
of  the  slide  at  O  will  appear  to  lie,  not  at  O,  but  on  the  backward  extension 
of  the  line  caf,  at  a.  If,  now,  the  tube  of  the  microscope  is  raised  and  focussed 
upon  a  mark  b  on  the  upper  surface,  the  amount  of  elevation  is  not  Ob,  the 
true  thickness  of  the  slide,  but  ab. 

Let  M  =  ab,  the  measured  thickness  of  the  mineral, 
D  =  Ob  =Jc,  the  actual  thickness, 
fco  =  i,  the  angle  of  incidence, 
f'ca'  =  c'ca  —  r,  the  angle  of  refraction. 

1  Le  Due  de  Chaulnes:  Sur  quelques  experiences  relatives  d  la  dioptique.  Hiltoire  de 
1'Academie  Royale  des  Sciences,  1767.  Paris,  1770,  162-175. 

Idem:  Memoir  e  sur  quelques  experiences  relatives  a  la  dioptique.  M6m.  de  1'Acad. 
France,  Ann6e  1767,  Paris  1770,  423-470.  In  particular  pages  430-435. 


ART.  208]  OBSERVATIONS  BY  ORDINARY  LIGHT  239 

Of 

tan  i     fc 
tan  r     ac' 


— / 

cc 


But  Of=ac',  and  cc'  =  ab  =  M,  wherefore 

tan  i  _cc'  __M 
tan  r     fc      D 

In  the  small  angles  here  used,  where  i  and  r  approach  o°,  the  tangent 
approaches  the  sine,  and  the  latter  may  be  substituted  in  the  equation, 
whereby 

sin  i     M 


sin  r     D 


(i) 


But  ^  —  =  -,  when  light  passes  from  a  denser  to  a  rarer  medium,  there- 
fore 

M     i  D 


That  is,  the  index  of  refraction  of  the  substance  is  equal  to  the  value  of  the 
true  thickness  divided  by  the  measured  thickness.  For  example:  By  the 
micrometer  screw  on  the  microscope  the  apparent  thickness  of  a  basal  sec- 
tion of  quartz  was  found  to  be  0.5  mm.,  by  actual  measurement  it  was  found 

to  be  0.77.     The  index  of  refraction,  therefore,  was  — —  —n=  1.54. 

The  weakness  of  the  method  lies  in  the  uncertainty  of  the  position  of 
sharpest  focus  and  inaccuracy  in  the  micrometer  reading,  a  difference  of 

±0.001  mm.  in  each  would  give,  in  the  above  example,  — =  1.535,  are- 
suit  decidedly  different  even  though  a  section  half  a  millimeter  in  thickness 
was  used.  If  the  section  were  of  the  thickness  of  a  normal  rock  slice,  the 
error  would  be  much  greater.  Another  error  is  caused  by  lost  motion  in 
the  micrometer  screw,  and  a  third  by  the  fact  that  the  section  may  not  be  of 
the  same  thickness  throughout  and  the  measurements  may  not  be  made  at 
the  exact  spot  where  the  indices  of  refraction  are  determined.  Fairly  ac- 
curate results  may  be  obtained  if  the  precaution  is  taken  to  avoid  lost  motion 
by  screwing  the  fine  adjustment  in  one  direction  only,  in  reading  the  top  and 
bottom  of  the  slice,  and  further  that  of  taking  a  large  number  of  readings  for 
thickness  at  various  places  in  the  mineral,  and  averaging  the  results. 

The  measurement  of  the  actual  thickness  of  a  plane  parallel  but  unmounted 
mineral  slice  may  be  made  by  placing,  upon  the  stage  of  the  microscope,  a 
glass  plate  having  a  reference  mark  upon  its  upper  surface,  and  sharply 


240 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  208 


focussing  upon  it. 


FIG.  345. — Micrometer  calipers.      (Central   Scientific 
Co.,    Chicago.) 


The  mineral  to  be  measured  is  then  placed  above  the 
mark  by  sliding  it  over  to  exclude 
the  air,  and  its  upper  surface  is 
brought  into  focus.  The  difference 
in  the  readings  of  the  microm- 
eter screw  of  the  fine  adjust- 
ment is  the  true  thickness. 
Another  method  of  measuring 
thickness  is  to  use  an  ordinary 
micrometer  screw  (Fig.  345)  or 
an  interference  sphaerometer1 
(Fig.  346).  The  latter  has  a  scale 

d  divided  into  0.5  mm.  spaces  and  a  disk  c  with  250  divisions,  permitting  a 
reading  to  0.002  mm. 
and  an  estimate  to 
o.ooi  mm.  The  instru- 
ment is  not  dependent 
upon  the  feeling  of  con- 
tact, as  are  ordinary 
micrometer  screws,  and 
it  is,  consequently, 
much  more  accurate. 
The  substance  to  be 
measured  is  placed 
upon  the  glass  plate  e, 
which,  in  turn,  rests 
upon  a  black  glass  plate 
/.  A  sodium  light  is 
placed  beyond  the  in- 
strument, and  the  in- 
stant the  rounded  end  of  the  screw  touches  the  substance  to  be  measured, 
^  interference  bands  appear  to  move  at  the  contact 
between  the  two  glasses  (e  and/). 

The  thickness  of  a  doubly  refracting  mineral 
sometimes  may  be  determined  by  means  of  its  bire- 
fringence (Art.  301). 

In  determining  the  index  of  refraction  of  a  thin 
section  of  a  mineral,  the  cover-glass  should  be  re- 
moved. If  this  is  not  done  a  correction  must  be 
applied  for  the  combined  cover-glass  and  balsam 
film.  A  value  of  1.52  may  be  taken  as  a  fair 
average  of  the  indices  of  glass  and  Canada  balsam, 

1  C.  Leiss:  Mittheilungen  aus  der  R.  Fuess'schen  Werkstatte.     Interferenz-Spharometer 
zur  genauen  Messung  der  Dicke  von  Kristallplatten.     Neues  Jahrb.,  1898  (II),  72-73. ' 


FIG.  346. — Interference  sphaerometer.     3/5   natural   size.     (Fuess.) 


FlG.  347. — Diagram  show- 
ing correction  to  be  applied 
for  cover-glass  in  measuring 
the  index  of  refraction  of  a 
substance  by  the  method  of 
the  Due  de  Chaulnes. 


ART.  210]  OBSERVATIONS  BY  ORDINARY  LIGHT  241 

and  since  D'  =  i.$2M',  instead  of  D'  =  Mf  as  it  would  in  air,  0.52  M'  must  be 
deducted  from  both  D  and  M  in  formula  (2),  M' being  the  measured  distance 
between  the  upper  surface  of  the  mineral  and  the  upper  surface  of  the  cover- 
glass,  D  the  true  thickness  of  the  mineral  and  the  cover-glass,  and  M  the 
apparent  thickness  of  the  crystal  plate  measured  from  its  bottom  to  its  top 

surface  (Fig.  347).    The  formula  becomes  w  =  -^—         ~^rl 

Various  modifications  of  the  method  of  the  Due  de  Chaulnes  have  been 
proposed  in  order  to  overcome  the  error  produced  by  slight  inaccuracies  in 
measuring  the  true  and  the  apparent  thickness. 

PROBLEM 

Determine,  by  the  method  of  the  Due  de  Chaulnes,  the  index  of  refraction  of  a  cleavage 
plate  of  fluorite,  about  0.5  mm.  in  thickness,  first  measuring  the  true  thickness  by  means  of 
the  fine  adjustment  of  the  microscope. 

209.  Brewster's  Method  for  Determining  the  Refractive  Index  of  a 
Liquid  (1813). — Sir  David  Brewster2  determined,  microscopically,  the  indices 
of  refraction  of  fluids  by  placing,  successively,  two  liquids  in  a  glass  trough 
with  a  perfectly  flat  bottom.  Let  n  be  the  index  of  refraction  of  a  known 
liquid  and  n'  that  of  the  one  td  be  determined.  If  D,  d,  and  df  are  the  dis- 
tances, measured  from  the  objective,  to  the  upper  surface  of  the  glass  bottom 
of  the  containing  vessel  through  air  only,  through  the  known  liquid,  and 
through  the  unknown,  then 

i  _i        _D 
n-i     D     ~d  ~d 


ri-i      I^_TL  D 

D~  d'          d' 

210.  Becquerel  and  Cahours'  Method  for  Determining  the  Refractive 
Index  of  a  Liquid  (1840). — Becquerel  and  Cahours3  used  a  similar  method, 
but  instead  of  measuring  D,  d,  and  d' ',  they  determined  the  number  of  divi- 
sions of  a  micrometer  (P,  p,  and  //,)  which  were  included  between  two  fixed 
lines  in  a  micrometer  ocular  on  examining  different  media.  These  values, 
as  may  easily  be  proved,  are  proportional  to  those  given  by  Brewster,  so  that 

1  See  Art.  152,  supra. 

2  David  Brewster:  A  treatise  on  new  philosophical  instruments,  Chapter  II,  Book  IV. 
Description  of  an  instrument  for  measuring  'he  refractive  powers  of  fluids,  and  of  a  method  of 
determining  the  refractive  powers  of  solids;  with  tables  of  the  refractive  powers  of  various  sub- 
stances.    Edinburgh,  1813,  240-288. 

3  Edmond  Becquerel  et  Auguste  Cahours:   Recherches  stir  les  pouvoirs  refringents  des 
liquides.     Comptes  Rendus,  XI  (1840,  Paris,  1841),  867-871. 

Abstract:  Untersuchungen  iiber  das  Brechvennogen  einiger  Fliissigkeiten.  Pogg.  Ann., 
LI  (XXI,  2nd  series),  1840,  427-433. 

16 


242  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  211 


_     . 

p 


The  standard  used  for  comparison  was  distilled  water  whose  mean  index 
was  taken  as  1.333. 

The  index  may  be  determined  directly  if  a  shallow  tray  of  the  liquid  is 
inserted  between  the  objective  and  a  reference  mark  on  a  glass  on  the  stage. 
If  d  represents  the  amount  which  the  objective  must  be  raised,  and  D  the 
depth  of  the  liquid  screen,  we  have 


of?) 


D 

or  n  = 


D-d 
But  D—d=M  (Fig.  344),  and  the  equation  becomes  n  =  -^,  as  before.    Bec- 

querel  and  Cahours  say  further,  that  "this  very  simple  formula  may  also 
be  used  to  determine  directly  the  index  of  refraction  of  a  solid,"  which  is, 
then,  of  course,  the  method  of  the  Due  de  Chaulnes. 

211.  Bertin's  Method  (1849).— In  the  method  of  Bertin,1  which  is 
applicable  to  solids,  no  micrometer  screw  is  necessary  on  the  microscope, 
but  the  measurements  are  made  with  stationary  objective  and  movable 
ocular.  A  finely  divided  glass  scale  is  placed  upon  the  upper  surface  of  the 
mineral  whose  index  of  refraction  is  to  be  measured,  the  tube  of  the  micro- 
scope is  drawn  out  to  its  full  extent,  and  the  enlargement  of  the  image  of  the 
scale  is  determined.  Let  G  be  its  value.  If  the  micrometer  is  now  placed 
beneath  the  mineral,  it  will  be  found  that  the  divisions  are  indistinct.  With- 
out changing  the  position  of  the  objective,  it  will  be  found  that  by  depressing 
the  ocular  (shortening  the  tube  length)  the  micrometer  may  again  be  brought 
into  focus,  but  the  enlargement,  in  this  case,  differs  from  that  first  determined. 
Let  7  be  the  new  value.  If  the  mineral  is  now  entirely  removed  from  the 
stage  and  the  micrometer  viewed  through  the  microscope  with  air  as  the 
only  intervening  medium,  a  farther  depression  of  the  ocular  is  necessary,  and 
a  third  enlargement  g  results. 

From  the  general  equation  of  lenses  (Eq.  9,  Art.  85)  we  have 

-1--     --+1- 

r    /i  A 

In  a  biconvex  lens,/!  will  be  negative  and/'2  positive,  and  our  equation  becomes 

III  ,/l/l  (      s 

7=71+A'ori+AT' 

1  A.  Bertin:  Sur  la  mesure  des  indices  de  refraction  des  lames  trans  par  entes  et  des  liquidcs 
a  I'aide  du  microscope  ordinaire.  Ann.  Chim.  et  Phys.,  XXVI  (1849),  288-296. 

Review:  Messung  der  Brechungsindexe  von  durchsichtigen  Flatten  mittelst  des  gewohn- 
lichen  Mikroskops.  Pogg.  Ann.,  LXXVI  (1849)  (XVI,  3d  series),  611-612. 


ART.  211] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


243 


\vhere/  is  the  principal  focus  of  the  system,/i  the  distance  of  the  object,  and  f'2 
the  distance  of  the  image  from  the  lens.  The  magnification  is  expressed  by  the 
ratio  of  the  size  of  the  image  to  that  of  the  object,  which  is  equal  to  the  ratio  of 
f*  to  /i,  whereby 


Substitute  in  equation  (i) 


7v 


(2) 


When  the  microscope  was  placed  below  the  mineral  section,  the  apparent  dis- 
tance of  the  object  from  the  lens  (Fig.  348)  was/i+M,  therefore 


When  the  mineral  section  was  removed  from  the 
stage,  the  distance  was/i+Z>,  and 


Subtracting  (^)  from  (3), 


FIG.  348. 


(5) 


y     G-  f 
Subtracting  (2)  from  (4) 

«"c  =  7" 

Dividing  (6)  by  (5)  and  combining  with  (2),  Art.  208, 

i      i      D 

i_2=^  =  M  =  n' 
r   G     f 

Simplifying,  we  have,  as  the  index  of  refraction, 


Gg-gy 


(6) 


(7) 


(8) 


In  determining  the  index  of  refraction  by  this  method,  an  object  microm- 
eter on  very  thin  glass  should  be  used.  It  should  be  placed  with  the  en- 
graved side  down  to  determine  G,  and  up  to  determine  7  and  g. 

Bertin  suggests  that  if  a  very  thick  plate  is  to  be  measured,  it  fs  better  to 
compare  it  with  a  plate  of  known  thickness  and  index  by  the  formula 


oft' -4)    '-' 

V nj    =g y 

~n)      g~y' 


(9) 


244  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  212 


This  equation  is  derived  from  those  preceding  as  follows: 
Subtracting  (3)  from  (4)  we  have 

i  _  i  _  D-M 

g      y~      f 

But  M  =  —  (Eq.  2,  Art.  208),  whereby  (10)  becomes 
n 


do) 


(n) 


g      v  f 

For  another  substance  with  measurements  Df  and  7',  and  index  «',  we  obtain 


«  V  / 

Dividing  (n)  by  (12),  we  obtain  equation  (9). 

PROBLEM 

Check,  by  Bertin's  method,  the  index  of  refraction  of  the  fluorite  plate  used  in 
the  previous  problem. 

212.  Sorby's  Method.  —  By  a  modification  of  de  Chaulnes'  method,  Sorby1  was 
enabled  to  measure  not  only  the  single  refractive  index  of  isotropic  substances,  but 
the  two  different  indices  of  those  that  are  anisotropic,  as  well.  To  make  his  deter- 
minations he  equipped  his  microscope  with  a  scale  and  vernier  whereby  he  was  able 
to  read  the  vertical  movement  of  the  tube  to  o.ooi  in.  (0.025  mm.).  In  modern 
microscopes  such  measurements  may  be  made  by  means  of  the  fine  focussing  adjust- 
ment which,  in  some  instruments,  give  readings  to  0.0005  mm«  Underneath  the 
stage,  and  as  far  below  the  lenses  of  an  achromatic  condenser  as  possible,  was  placed 
a  glass  plate  (Fig.  349)  upon  which  were  engraved  two  sets  of  fine  lines.  These 
were  ruled  in  two  directions  at  right  angles  to  each  other  and  o.oi  in.  (0.254  mm.) 
apart.  The  lines  of  this  grating  could  be  brought  to  a  focus  by  means  of  the  con- 
denser, either  upon  the  lower  or  upper  surface  of  the  specimen  or  anywhere 
within  it,  and  appeared  there  as  a  much  reduced  image.  Close  to  the  glass  grating 
was  an  iris  diaphragm  whereby  a  circular  image  of  any  diameter  could  be  brought 
into  focus  in  the  same  plane  as  the  ruled  lines.  Below  the  diaphragm  was  a  nicol 

1  H.  C.  Sorby:  On  a  simple  method  for  determining  the  index  of  refraction  of  small  por- 
tions of  transparent  minerals.  Preliminary  notice.  Mineralog.  Mag.,  I  (1877),  97-98. 

Idem:  President's  Address,  Mineralogical  Society.     Ibidem,  193-208. 

Idem:  On  some  hitherto  undescribed  optical  properties  of  doubly  refracting  crystals.  Pre- 
liminary notice.  Proc.  Roy.  Soc.  London,  XXVI  (1877),  384-386. 

Idem:  On  the  determination  of  the  minerals  in  thin  sections  of  rocks  by  means  oj  their 
indices  of  refraction.  Mineralog.  Mag.,  II  (1878),  1-4. 

Idem:  Further  improvements  in  studying  the  optical  characters  of  minerals.  Ibidem,  II 
(1878),  103-105. 

Idem:  On  a  new  method  for  studying  the  optical  properties  of  crystals.  Ibidem,  XV 
(1909),  189-215. 


ART.  212]  OBSERVATIONS  BY  ORDINARY  LIGHT  245 

prism  and  another  was  above  the  eyepiece,  and  either  or  both  could  be  rotated  or 
thrown  out  of  position.  A  2-in.  (50  mm.)  eyepiece  and  a  2/3-in.  (16.9  mm.)  objec- 
tive were  used,  the  latter  stopped  down  to  a  13°  aperture  by  means  of  a  cap  with  a 
small  opening.  Another  cap,  with  a  semi-circular  opening  cutting  off  exactly  one- 
half  of  the  front  lens  in  any  desired  direction,  was  used  to  determine  the  plane  of 
polarization  of  any  beam  that  had  passed  through  the  mineral  under  examination. 
Determinations  were  made  both  on  mineral  sections  cut  with  plane-parallel 
faces  and  on  natural  crystals,  the  latter  possessing  the  advantage  of  having  opposite 
faces  truly  parallel.  If  the  surfaces  were  rough,  a  drop  of  oil,  of  approximately  the 
same  index  as  the  mineral,  was  placed  above  and  below  it,  and  protected  by  a  cover- 
glass.  This  gave  rise  to  a  small  error,  but  with  a  specimen  from  i/io  to  1/2  in.  in 
thickness,  it  was  of  no  great  moment. 


FIG.  349.  FIG.  350.  FIG.  351-     FIG.  352.        FIG.  353-          FIG.  354- 
FIGS.  349  TO  354.  —  Images  seen  through  mineral  plates  by  the  method  of  Sorby. 

With  the  microscope  so  arranged,  the  phenomena  observed  are  as  follows: 
Isotropic  Substances.  —  On  looking  at  the  image  of  the  grating  without  any  inter- 
vening object,  both  sets  of  lines  are  seen  at  the  same  focus,  as  shown  in  Fig.  349.  l 
If  an  isotropic  mineral  or  a  transparent  amorphous  body  with  plane-parallel  faces 
is  placed  on  the  stage,  the  two  sets  of  lines  can  still  be  seen  in  one  plane  although 
at  a  different  focus  than  before.  No  matter  how  much  the  stage  is  rotated,  the 
lines  remain  in  view  and  the  circle  is  not  distorted.  Isotropic  substances,  conse- 
quently, have  no  special  focal  axis.  They  are  also  unifocal  because  all  parts  of  the 
only  image  lie  at  the  same  focal  distance. 

The  index  of  refraction  of  an  isotropic  substance,  consequently,  is  determined, 
as  explained  above,  by  the  formula 

D 


Anisotropic  Crystals.  —  The  phenomena  observed  in  minerals  having  double 
refraction  are  totally  different,  and  in  order  to  examine  separately  the  two  rays, 
which  are  polarized  in  opposite  planes,  it  is  necessary  to  use  a  rotating  analyzer, 
either  within  the  tube  or  above  the  eyepiece,  and  so  turned  that  it  permits  either 
one  or  the  other  of  the  rays  to  pass  through.  In  every  case  it  will  be  found  that  the 
ordinary  ray  is  unifocal  and  acts  as  does  the  light  in  isotropic  substances. 

UNIAXIAL  CRYSTALS 

Section  cut  Perpendicular  to  the  Optic  Axis.  —  If  a  section  of  calcite,  0.25  in.  in 
thickness  and  cut  at  right  angles  to  the  optic  axis,  is  examined,  two  images  ap- 

1  For  photographic  reproductions  of  these  figures,  see  the  beautiful  illustrations  given 
by  Dr.  Hans  Hauswaldt:  Inter  jerenzerscheinungen  im  Polaris  irten  Licht,  3te  Reihe,  Magde- 
burg, 1908,  plates  35-36-37. 


246  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  212 

pear,  each  showing  both  sets  of  ruled  lines.  They  are  directly  superimposed  but  lie 
in  different  focal  planes,  as  though  there  were  two  sets  of  lines  ruled  on  opposite 
sides  of  a  glass  plate.  On  bringing  one  image  into  focus,  the  circle  appears  sharp 
and  undistorted,  but  the  other  image,  which  is  seen  out  of  focus,  appears  as  a  large 
blurred  circle  surrounding  the  first  (Fig.  352).  On  changing  the  focus,  the  second 
image  becomes  sharp  and  the  first  forms  the  blurred  halo.  Looking  straight  down, 
the  ordinary  cannot  be  distinguished  from  the  extraordinary  image,  but  if  the 
section  be  somewhat  inclined  the  images  separate  and  the  two  rays  may  be 
differentiated. 

Placing  the  semi-circular  stop  over  the  objective,  with  the  straight  cut  of  the 
•opening  parallel  to  one  of  the  sets  of  lines  in  the  grating,  produces  the  effect  of 
slightly  inclining  the  section  by  causing  the  light  to  pass  through  obliquely.  It 
thus  shows  the  ordinary  image  to  be  unifocal  and  the  extraordinary  image  to  be 
slightly  bifocal,  as  explained  below. 

The  index  of  refraction  of  the  ordinary  ray,  in  the  section  of  calcite  examined  by 
Sorby,  was  found  to  be  equal  to  1.659;  the  apparent,  but  not  the  true  value  for  the 
extraordinary,  1.335.  The  value  of  the  apparent  extraordinary  ray  in  various  direc- 
tions should  be,  according  to  Stokes,  l  equal  to  the  square  of  the  true  index  of  the 
extraordinary  ray  divided  by  the  true  index  of  the  ordinary.  In  this  case 


Section  Parallel  to  the  Cleavage.  —  The  images  of  the  circular  opening,  seen  through 
a  section  of  calcite  cut  parallel  to  the  cleavage,  appear  widely  separated  in  the  plane 
of  the  principal  axis  (Fig.  353),  and  lie  at  different  focal  distances.  The  image  due 
to  the  ordinary  ray  is  in  no  way  distorted  and  lies  in  the  center  of  the  field,  that  due 
to  the  extraordinary  ray  is  elongated  and  appears  to  lie  at  a  lower  level  and  to  one 
side.  It  will  be  found  that  there  is  no  single  adjustment  in  which  this  image  is 
completely  in  focus.  In  one  position  of  the  objective  it  appears  as  an  elongated 
band  with  two  sides  parallel  to  each  other  and  parallel  to  the  axis  of  the  crystal  and 
with  illy  defined  ends.  On  raising  the  tube  of  the  microscope,  the  band  changes 
into  a  poorly  defined  circle  several  times  larger  than  the  real  one,  and  then  into  a 
band  elongated  in  a  direction  perpendicular  to  the  former. 

With  the  analyzer  arranged  so  that  only  the  ordinary  image  appears,  it  will  be 
found  to  be  unifocal,  and  both  sets  of  lines  of  the  grating  will  appear,  no  matter 
what  the  azimuth  of  the  crystal.  The  index  of  refraction  was  found  by  Sorby  to 
be  1.657. 

When  the  crystal  is  so  turned  that  the  lines  of  the  grating  are  parallel  and  per- 
pendicular to  the  axis  of  the  crystal,  and  the  analyzer  so  arranged  that  only  the 
extraordinary  image  appears,  there  will  be  two  widely  separated  focal  points  at 
each  of  which  only  one  system  of  lines  can  be  seen.  That  at  which  the  lines  parallel 
to  the  axis  appear,  give  an  index  of  1.412,  while  that  at  which  those  perpendicular 
appear  give  approximately  1.578,  but  the  latter  are  poorly  defined  unless  light 
passed  through  red  glass  is  used.  The  extraordinary  image  is  therefore  truly 
bifocal. 

1  G.  G.  Stokes:  On  the  foci  of  lines  seen  through  a  crystalline  plate.  Proc.  Roy.  Soc., 
London,  XXVI  (1877),  386-401. 


ART.  212]  OBSERVATIONS  BY  ORDINARY  LIGHT  247 

In  sections  cut  at  greater  inclinations  with  the  axis,  the  bifocal  image  becomes 
more  and  more  nearly  unifocal  until  in  sections  perpendicular  to  the  axis,  it  is 
entirely  so,  as  explained  above. 

Section  Parallel  to  the  Principal  Axis. — On  examining  a  section  of  calcite,  0.2  in. 
in  thickness  and  parallel  to  the  optic  axis,  it  will  be  found,  when  the  analyzer  per- 
mits only  the  extraordinary  ray  to  pass  through,  that  there  are  two  different  foci 
at  which  the  lines  of  the  grating  are  visible.  The  circular  hole  is  elongated  first  in 
one  direction  (Fig.  350)  and  then  in  the  other  (Fig.  351),  and  in  each  case  only  one 
system  of  rulings  can  be  seen,  and  then  only  when  the  grating  is  so  arranged  that 
the  lines  are  parallel  and  perpendicular  to  the  axis  of  the  crystal.  The  image  of  the 
extraordinary  ray  is  bifocal,  and  since  the  rulings  disappear  when  the  stage  is 
rotated,  it  has  a  definite  focal  axis.  The  ordinary  ray  gives  an  image  not  distorted 
and  at  a  single  focus. 

The  index  of  refraction  of  the  ordinary  image  is  its  true  index.  That  for  the 
lines  parallel  to  the  principal  axis  of  the  crystal  is  the  true  index  of  the  extraordinary, 
while  that  of  the  lines  perpendicular  to  this  axis  is  the  apparent  index  and  is  equal, 
according  to  Stokes,  to  the  square  of  the  index  of  the  ordinary  ray  divided  by  that 
of  the  extraordinary,  in  this  case  producing  a  result  of  1.868,  which  is  greater  than 
that  of  the  ordinary  ray. 

Sorby,  in  regard  to  the  apparent  index,  says:  "The  phenomenon  seen  with  the 
microscope  depends  entirely  on  the  power  of  the  object  glass  to  collect  divergent 
rays.  In  the  case  of  substances  having  no  double  refraction,  this  divergence  merely 
obeys  the  laws  of  ordinary  refraction,  and  enables  us  to  measure  the  index  in  the 
manner  already  explained;  but  in  the  case  of  the  extraordinary  ray,  the  light  is  bent 
from  the  normal  line  unequally  and  in  opposite  directions,  and  may  thus  enter  the 
object  glass  at  an  angle  of  divergence  greater  or  less  than  that  depending  on  the 
index  of  refraction.1 

BIAXIAL  CRYSTALS 

Section  Perpendicular  to  the  Principal  Axis. — Crystals  of  aragonite  and  orpiment 
were  used.  The  circular  hole  of  the  diaphragm  appears  as  two  crosses  (Fig.  354) 
lying  in  widely  different  focal  planes,  each  cross  being  itself  bifocal  and  polarized 
in  opposite  planes.  There  may  thus  be  four  different  apparent  indices,  but  in 
sections  cut  in  particular  directions  one  or  two  pairs  may  become  equal  and  have 
the  appearance  of  a  unifocal  image,  differing,  however,  from  unifocal  images  due 
to  an  ordinary  ray,  in  becoming  bifocal  when  one-half  the  front  lens  of  the  objective 
is  covered  with  the  semi-circular  stop.  There  is  no  ordinary  ray. 

If  the  section  is  inclined  away  from  the  axis,  the  image  becomes  much  less  sym- 
metrical. 

Sections  Parallel  to  the  Principal  Axis. — Sections  parallel  to  the  principal  axis 
give  different  figures,  depending  upon  their  orientation  with  respect  to  the  other 
axes.  When  parallel  to  the  principal  and  to  one  of  the  secondary  axes,  a  cross  with 
unequal  arms,  at  four  different  foci,  is  obtained;  when  cut  parallel  to  the  principal 
and  along  the  diagonal  of  the  secondary  axis,  one  image  is  decidedly  bifocal  and  one 
unifocal.  The  latter,  however,  is  caused  by  an  extraordinary  ray,  as  may  be  shown 
by  passing  an  inclined  ray  through  it. 

1  H.  C.  Sorby:  Op.  cit.,  Mineralog.  Mag.,  I  (1877),  199. 


248  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  212 

Determination  of  Indices  of  Refraction. — In  determining  the  real  value  of  the 
indices  of  refraction,  the  following  facts  must  be  remembered. 

1.  A  crystal  having  no  double  refraction  has  no  bifocal  image,  and  its  index  of 
refraction  is  the  true  index. 

2.  The  ordinary  ray  of  a  uniaxial  crystal  gives  a  unifocal  image,  and  its  index  of 
refraction  is  its  true  index,  no  matter  what  may  be  the  orientation  of  the  section. 

3.  Biaxial  crystals  have  two  bifocal  images  whose  focal  axes  are  always  perpen- 
dicular to  the  plane  of  polarization  of  the  images.     In  any  bifocal  image  one  apparent 
index  is  true  when  the  corresponding  principal  focal  axis  is  parallel  to  the  plane  of 
the  section.     If,  therefore,  a  biaxial  crystal  is  cut  parallel  to  two  principal  axes, 
each  image  will  give  one  true  index;  the  third  may  be  calculated.     If  the  crystal 
is  cut  parallel  to  only  one  axis,  only  one  true  index  can  be  determined,  and  if  parallel 
to  no  axis,  none  of  the  true  indices  can  be  obtained. 


CHAPTER   XIV 
OBSERVATIONS  BY  ORDINARY  LIGHT  (Continued) 

DETERMINATION  OF  THE  REFRACTIVE  INDICES  OF  A  MINERAL  BY  THE 
IMMERSION  OR  EMBEDDING  METHOD 

213.  Maschke  (1872-1880). — If  a  crystal  is  immersed  in  a  liquid  of  a 
different  refractive  index,  and  is  examined  under  the  microscope,  it  will  be 
seen  that  its  borders  are  either  dark  or  colored,  due  to  the  reflection  of  the 
light  at  the  edges.  If  the  index  of  the  immersion  liquid  is  exactly  the  same 
as  that  of  the  mineral,  the  borders  are  lost  and,  if  the  mineral  is  colorless, 
the  latter  disappears  from  view.  This  fact  had  long  been  known  but  its 
applicability  to  the  separation  of  microscopic  mineral  fragments  appears 
first  to  have  been  recognized  by  Maschke1  in  1872,  while  engaged  in  a  study 
of  quartz  and  tridymite.  He  determined  the  fact  that  as  the  index  of  the 
immersion  fluid  approaches  that  of  the  mineral,  the  dark  borders  give  way  to 
colors,  which  he  ascribed  to  interference.  He  stated  that  when  the  index  of 
the  liquid  is  lower  than  that  of  the  mineral,  the  latter  appears  bluish  or 
bluish-green  with  a  reddish  rim,  and  that  when  the  index  of  the  liquid  is 
greater  than  that  of  the  mineral,  the  latter  appears  reddish  with  a  bluish  or 
bluish-green  rim.  He  also  suggested  that  just  as  we  now  have  a  scale  of 
hardness,  so  might  also  a  series  of  immersion  liquids  be  prepared  for  the 
comparison  of  refractive  indices.  He  proposed,  as  such,  cassia  oil,  tur- 
pentine, and  poppy  oil,  or  mixtures  of  these,  alcohol,  and  a  solution  of  mer- 
curic nitrate  of  various  degrees  of  dilution. 

In  a  later  paper,  Maschke2  correctly  recognized  the  colors  as  micro- 
prismatic,  and  indicated  how  they  might  be  brought  out  by  inclined  illumi- 
nation. To  produce  this  he  displaced  the  lower  diaphragm  laterally  or, 
more  simply,  fastened  across  the  front  lens  of  the  objective,  by  means  of  a 
touch  of  wax  on  either  side  of  the  casing,  a  thin,  dull-black  strip  of  paper, 
1.5  to  2  mm.  in  width  and  with  sharp  edges.  The  paper  was  pressed  into 
close  contact  with  the  lens  and,  since  a  low  power  was  used,  the  opening  was 
sufficiently  large.  The  diaphragm  was  now  closed  until  the  paper  appeared 
as  a  narrow,  black  bar  across  the  middle  of  the  field. 

For  the  measurement  of  the  indices  of  doubly  refracting  minerals,  Mas- 
chke made  use  of  a  polarizer,  and  determined  the  values  in  different  direc- 

1  O.  Maschke:  Ueber  Abscheidung  krystallisirter  Kieselsaure  aus  wdssrigen  Losungen. 
Pogg.  Ann.,  CXLV  (5  ser.  XXV,  1872),  549-578,  in  particular  568-569. 

2  O.  Maschke:  Ueber  eine  mikroprismatische  Methode  zur  Unterscheidung  fester  Sub- 
stanzen.    Wiedem.  Ann.,  N.  F.  XI  (1880),  722-734. 

249 


250 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  214 


tions.  Among  the  fluids  used  in  his  later  work  were  water,  amyl  alcohol, 
glycerine,  almond  oil,  and  cassia  oil,  the  latter  two  mixed  in  varying  propor- 
tions. He  thus  had  a  series  of  indicators  with  values  from  1.333  to  1-606. 

214.  Sorby  (1877). — No  further  use  was  made  of  the  immersion  method 
for  determining  the  relative  refractive  indices  of  a  fluid  and  a  solid  until  the 
method  was  rediscovered  in  1900  by  Schroeder  van  der  Kolk.1    The  method 
of  reducing  the  dark  borders  by  immersion  had  been  employed,  however,  and 
Sorby2  made  use  of  a  diaphragm  and  of  inclined  illumination.     In  his  Presi- 
dential address  to  the  Royal  Microscopical  Society,  in  1877,  he  called  atten- 
tion to  the  fact  that  when  a  mineral  is  immersed  in  a  fluid  having  a  refractive 
index  but  slightly  different,  no  outline  is  seen  if  the  angle  of  convergence  of 
the  light  is  considerable,  but  by  cutting  down  the  cone  of  light,  the  outlines 
become  more  and  more  distinct  and  the  shading  greater  and  greater.     He 
spoke  of  the  importance  of  having  the  means  of  varying  the  angle  of  devia- 
tion from  a  direct  line  by  means  of  a  diaphragm  below  the  condenser. 

215.  Thoulet  (1870). — In  1879,  Thoulet3  described  a  heavy  solution  of 
potassium  mercuric  iodide,  previously  used  by  Sonstadt  but  now  generally 
known  as  Thoulet 's  solution,4  for  determining  specific  gravities.     It  has  a 
very  high  index  of  refraction,  the  maximum  being  1.7333  for  sodium  light. 
Being  miscible  with  water  in  all  proportions,  a  range  of  indices  from  1.333  to 
1.733  m&y  be  obtained.     Goldschmidt5  computed  the  values  given  below, 
for  sodium  light  and  at  18°  C. 

TABLE  SHOWING  THE  RELATIONS  BETWEEN  SPECIFIC  GRAVITY  AND  REFRACTIVE 
INDEX  OF  THOULET'S  SOLUTION  IN  SODIUM  LIGHT  AND  AT  18°  C. 


Specific 
gravity 

HD 

Specific 
gravity 

HD 

Specific 
!     gravity 

HD       ' 

Specific 
gravity 

UD 

3-2 

3-i 
3-o 

2    Q 

•7333 
•7145 
.6956 
6768 

2.7 

2.6 

2-5 
2    4. 

•6395    | 
.6207 
.6020 
r8?2 

2  .  2 
2  .  I 
2  .0 
I    0 

-5457 
.5270 
.5090 

4-QIO 

i-7 
1.6 

i-5 

I-4551 
I-437I 
1.4186 

2.8 

.6582   j 

2.3 

•5645 

1.8 

•  4731 

i  .0 

1-3333 

In  1880,  Thoulet6  used  the  immersion  method,  to  a  certain  extent,  and 

1  See  footnote  24,  Art.  228. 

2  H.  C.  Sorby:  Anniversary  Address  of  the  President  of  the  Royal  Microscopical  Society. 
Mon.  Microsc.  Jour.,  XVII  (1877),  1.17-118. 

3  J.  Thoulet:  Separation  mechanique  des  divers  elements  miner alogiques  des  roches.     Bull. 
Soc.  Min.  France,  II  (1879),  17-24« 

4  For  the  method  of  preparation  see  Art.  454.     The  solution  is  decomposed  by  metallic 
iron.     It  is  also  extremely  poisonous  and  should  be  used  with  great  caution. 

6  V.  Goldschmidt:  Ueber  V erwendbarkeit  einer  Kaliumquecksilberjodlosung  bet  miner al- 
ogischen  und  petrographischen  Untersuchungen.  Neues  Jahrb.,  B.  B.,  I  (1881),  179-238. 

CJ.  Thoulet:  De  Vapparence  dite  chagrinee  presentee  par  im  nombre  de  miner aux  ex- 
amines en  lames  minces.  Bull.  Soc.  Min.  France,  III  (1880),  62-68. 


ART.  219] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


251 


mentioned  that  the  shagreen  surface  of  minerals  disappears  when  the  refrac- 
tive indices  of  fluid  and  solid  are  the  same.  He  used  water,  alcohol,  gly- 
cerine, olive  oil,  beech-nut  oil,  clove  oil,  cinnamon  oil,  bitter-almond  oil,  and 
bisulphide  of  carbon. 

216.  Stephenson  (1880). — In  order  to  obtain  relief  in  biologic  specimens, 
Stephenson,1  in  1880,  immersed  them  in  phosphorus,  bisulphide  of  carbon, 
or.  solutions  of  sulphur      He  gave  a  table  of  the  refractive  indices  of  various 
immersion  substances  but  made  no  attempt  to  determine  the  index  of  the 
embedded  material. 

217.  Rohrbach    (1883). — Rohrbach,2   in    1883,  proposed  a  solution  of 
barium  mercuric  iodide,  now  generally  known  as  Rohrbach's  solution,  for 
the  determination  of  the  specific  gravity  of  minerals,  and  as  one  having  a 
high  refractive  index.     He  gives  1.7932  to  1.7928  at  23°  C.  and  in  sodium 
light.     The  relation  between  specific  gravity  and  refractive  index  is  shown 
in  Fig  722. 

218.  Brauns    (1886). — Methylene    iodide,    introduced    by   Brauns3   in 
1886  as  a  heavy  solution  and  as  one  having  a  high  index  of  refraction,  is  a 
light  yellow  fluid,  unaltered  by  contact  with  air  and  miscible  in  all  propor- 
tions with  benzol.     Undiluted,  ks  indices  of  refraction  at  different  tempera- 
tures and  by  sodium  light  are  as  follows: 

REFRACTIVE  INDICES  OF  METHYLENE  IODIDE  AT  DIFFERENT  TEMPERATURES  BY 

SODIUM  LIGHT 


Temp. 

nD 

Temp. 

nD 

•   Temp. 

nD 

Temp. 

nD 

£ 

E 

.74873 
.  74802 

•74731 
.  74660 

11° 

12° 

? 

•  74447 
.74376 

•  74305 
74234 

$ 

£ 

.74021 
•73950 
.73879 
73808 

IO  K>  K) 

C/i  .£>.  Oo 

000 

i  •  73595 
I-73524 
i  •  73453 

0° 

.  74589 

4° 

74163 

21° 

.  7^737 

10° 

I-745I8 

16° 

.74092 

22° 

.73666  1 

31° 

1.7300 

219.  Bertrand  (1888). — Bertrand4  increased  the  index  of  refraction  of 
methylene  iodide  by  dissolving  in  it  a  large  quantity  of  sulphur  by  means 

1  J.  W.  Stephenson:  On  the  visibility  of  minute  objects  mounted  in  phosphorus,  solutions 
of  sulphur,  bisulphide  of  carbon  and  other  media.    Jour.  Roy.  Microsc.  Soc.,  Ill  (1880), 
564-567. 

2  Carl  Rohrbach:    Ueber  eine  neue  Fltissigkeit  von  hohem  specifischen  Gewicht,  hohem 
Brechungsexponenten  und  grosser  Dispersion.     Wiedem.  Ann.,  N.  F.,  XX  (1883),  169-174. 

For  the  method  of  preparation  see  Art.  456. 

3  R.  Brauns:    Ueber  die  V erwendbarkeit  des  Methylenjodids  bei  petrographischen  und 
optischen  Untersuchungen.     Neues  Jahrb.,  1886  (II),  72-78. 

For  the  method  of  use,  see  Art.  457. 

4  Emile  Bertrand:    Liquides  d' indices  superieurs  a   1.8.     Bull.  Soc.  Min.  France,  XI 
(1888),  31. 


252  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  220 

of  heat.  On  cooling,  large  crystals  of  sulphur  were  formed,  leaving  a  liquid 
having  a  refractive  index  above  1.8.  On  dissolving  iodine  and  sulphur  in 
methylene  iodide,  a  liquid  having  an  index  greater  than  1.85  was  obtained. 
The  proportions  of  iodine  and  sulphur,  in  the  latter  liquid,  are  not  given  by 
Bertrand,  and  the  writer  has  been  unable  to  obtain  a  higher  refractive  index 
than  1.82  after  the  fluid  becomes  cold. 

220.  Klein  (1890). — Klein,1  in  1890,  1891,  and  later,  used  the  immersion 
method  to  get  rid  of  the  boundaries  of  crystals  in  making  various  examina- 
tions under  the  microscope,  such  as  extinction,  optic  angles,  and  so  on,  but 
he  did  not  specifically  apply  it  to  the  determination  of  refractive  indices, 

221.  Schroeder  van  der  Kolk  (1892). — Schroeder  van   der   Kolk,2   in 
1892,  used  inclined  illumination  to  bring  out  certain  properties  of  minerals, 
but  as  yet  had  not  applied  it  to  the  determination  of  their  indices. 

222.  Zirkel  (1893). — Zirkel,3  in  1893,  gave  a  list  of  twenty-six  immersion 
fluids,  but  suggested  no  way  by  which  to  determine  whether  fluid  or  solid  has 
the  greater  index.     The  accuracy  of  some  of  the  higher  indices  is  questioned. 
No  references  are  given  to  the  authority  for  the  data. 

223.  Retgers  (1893). — Retgers,4  in  1893,  proposed  phosphorus  in  a  molten 
condition  or  as  a  concentrated  solution  in  carbon  bisulphide  as  a  medium  of 
high  refractive  index.     A  grain  of  colorless  to  yellow  phosphorus,  the  size 
of  a  pin  head,  is  rapidly  dried  with  a  piece  of  linen  or  filter  paper,  and  is 
placed  on  the  object-slide  and  quickly  covered  with  a  cover-glass.     Upon 
heating,  high    up   over  a   small  naked  flame,  the  phosphorus  melts,  and, 
if  the  precaution  is  taken  to  press  down  firmly  upon  the  cover-glass,  it  will 
spread  out  into  a  flat  drop,  i  or  2  cm.  in  diameter.     There  is  no  danger  of 
ignition  since  no  air  is  admitted.     Even  if  a  small  quantity  is  squeezed  out 
beyond  the  cover-glass  and  ignites,  it  burns  out  without  igniting  the  part 
covered  from  the  air.     The  phosphorus  should  not  be  heated  above   the 
melting-point  (44°  C.)  or  it  will  turn  dark  yellow  or  red.     After  the  phos- 
phorus is  fluid,  it  will  remain  so  for  a  considerable  time  and  have  an  index  of 

1  Carl  Klein:  Ueber  eine  Methode,  game  Krystalle  oder  Bruchstiicke  ders'lben  zu  Untcr- 
suchungen  im  parallelen  und  im  convergenten  polarisirten  Lichte  zu  verwenden.  Sitzb.  Akad. 
Wiss.  Berlin,  1890  (I),  347-352. 

Idem:  Ueber  die  Methode  der  Einhiillung  der  Krystalle  zum  Zweckihrer  optischen  Erfor- 
schung  in  Medien  gleicher  Brechbarkeit.  Reprinted,  with  additions  by  the  author,  from 
Stizb.  Akad.  Wiss.  Berlin,  1890,  703,  in  Neues  Jahrb.,  1891  (I),  70-76. 

2J.  L.  C.  Schroeder  van  der  Kolk:  Ueber  die  Vortheile  schiefer  Beleuchtung  bei  der 
Untersuchung  von  Diinnschlijfen  im  parallelen  polarisirten  Lichte.  Zeitschr.  f .  wiss.  Mikrosk. 
VIII  (1891-2),  456-8. 

8  F.  Zirkel:    Lehrbuch  der  Petrographie,  I.  2te.  Aufl.,  Leipzig,  1893,  40. 

4  J.  W.  Retgers :  Der  Phosphor  als  stark  lichtbrechendes  Medium  zu  petrographischen 
Zwecken.  Neues  Jahrb.,  1893  (II),  130-134,  and  correction,  Ibidem,  1894  (I),  424. 


ART.  224]  OBSERVATIONS  BY  ORDINARY  LIGHT  253 

refraction  in  sodium  light  of  2.075.  On  cooling,  the  phosphorus  remains 
perfectly  clear  and  will  not  form  a  crystalline  aggregate  although  it  is  iso- 
metric. Its  index  of  refraction  is  2.144  by  sodium  light.  After  having  made 
a  determination,  object  and  cover -glass  may  be  freed  from  phosphorus 
by  dipping  them  into  nitric  acid,  which  will  reduce  the  phosphorus  to  phos- 
phoric acid. 

Dissolved  in  carbon  bisulphide,  phosphorus  is  in  no  danger  of  ignition  if 
properly  used,  nor  does  it  oxidize  into  the  red  or  opaque  form.  It  should 
not,  however,  be  kept  in  stock,  but  one  should  proceed  as  follows:  A  grain 
of  the  mineral  to  be  examined  is  placed  on  the  object-slide,  and  with  it  a  piece 
of  phosphorus  about  i  mm.  in  diameter.  It  is  then  quickly  covered  with  a 
cover-glass,  and  one  or  two  drops  of  carbon  bisulphide  are  permitted  to  flow 
beneath  the  edge,  pressure  being  applied  at  the  same  time  to  the  cover- 
glass.  The  phosphorus  soon  dissolves  and  is  much  more  transparent  than 
in  the  molten  state,  although  its  index  of  refraction  is  only  about  1.95  (P, 
n  =  2.i4,  CSz,  #  =  1.63)  at  room  temperature.  Object  and  cover-glass  may 
be  cleaned  by  dipping  in  carbon  bisulphide. 

224.  Ambronn  (1893). — By  the  methods  used  before  1893,  the  process 
of  finding  immersion  fluids  of  the  proper  indices  was  extremely  tedious, 
especially  when  working  with  anisotropic  minerals  for  which  it  is  necessary 
to  make  observations  above  a  nicol  prism,  placing  the  plane  of  polarization 
parallel  first  to  one  and  then  to  another  vibration  direction,  and  selecting 
refractive  fluids  corresponding  to  each.  Ambronn1  said  that  it  is  much 
easier  to  find  a  fluid  with  an  index  of  refraction  intermediate  between  the 
indices  in  two  directions  at  right  angles  to  each  other  in  the  mineral,  than  to 
find  two  that  exactly  coincide.  In  such  a  fluid  the  boundaries  of  the  mineral 
do  not  disappear  unless  the  stage  is  rotated  to  a  particular  position  with 
reference  to  the  direction  of  vibration  of  the  polarizer.  If,  then,  one  deter- 
mines the  amount  of  rotation  in  azimuth  necessary  in  each  of  two  such 
intermediate  fluids  with  different  indices,  he  can  determine  the  indices  of 
refraction  of  the  mineral  from  the  equations: 

«22  COS2  (f>\  —  Hi2  COS2  <p2 

o>i2  =  -      — ir- 

-I    tn*    


COS     (pi  —  COS     <f>2 

w22  cos2  (f>i  —  ni2  sin2 


sm    <pi  —  sm    <f>z 

in  which  o>i  and  ei  are  the  indices  of  refraction  to  be  determined  in  two 
directions  at  right  angles  to  each  other,  n\  the  index  of  refraction  of  the  first 
immersion  liquid,  «2  that  of  the  second  liquid,  <pi  the  angle  between  c  and  the 

1  H.  Ambronn:  Ueber  eine  neue  Methode  zur  Bestimmung  der  Brechungsexponenten 
anisotroper  mikroskopischer  Objecte.  Ber.  Gesell.  Wiss.,  Leipzig.,  Math.-phys.  Kl.,  XLV 
(1893),  316-318. 


254 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  225 


position  where  the  boundaries  of  the  mineral  disappear  in  the  first  liquid, 
and  <f>z  the  angle  between  c  and  the  position  of  disappearance  in  the  second. 
This  method  is  applicable  only  to  very  thin  sections  of  minerals,  an  appreci- 
able error  arising  if  they  are  thick.1  The  accuracy  of  the  method  does  not 
appear  to  be  very  great,  Ambronn's  results  varying  in  the  second  decimal 
place. 

225.  Ambronn  (1896). — In  a  later  paper,  Ambronn2  called  attention 
to  the  colored  borders  seen  at  the  contact  between  a  mineral  and  an  im- 
mersion fluid  when  the  indices  of  refraction  differ  but  slightly,  say  in  the 
third  decimal  place.  As  an  example  he  gives  the  contact  between  glass  and 
a  mixture  of  monobromnaphthylene  and  xylol.  The  indices  of  the  two  for 
different  rays  are 


Bine 


Red 


Glass 

Fluid 

a 

nn 

I     ^I  34. 

i   ^007 

4°       0' 

n  ~.  .  . 

I    5144 

i  5116 

s°  30' 

c  

nD  

I.5I70 

1.5170 

0°        0' 

UE  

1.5204 

1.5236 

3°   V 

n^  .  . 

I  .  5234 

i  .  5296 

5°    10' 

F 

n^ 

I     52QO 

i   ^406 

7°      o' 

G  

nH  

1-5335 

1.5500 

8°   20' 

FIG.  355. — The  cause  of  colored 
borders  around  minerals. 


Consequently,  when  white  light  is  used  (Fig.  355),  since  the  dispersion  for 
fluids,  in  general,  is  greater  than  for  solids,  the  yellow  rays  (n^)  will  pass 
through  both  media  without  change  of  direction,  the  red  will  be  bent  toward 
the  glass,  and  the  blue  toward  the  liquid.3  Ambronn  suggests,  for  the  de- 
termination of  refractive  indices,  that  a  series  of  refractive  fluids,  differing 
by  0.005,  be  prepared.  When  the  color  effects  are  well  marked,  the  refractive 
index  for  the  yellow  cannot  differ  greatly  in  the  two  media.  The  observa- 
tion may  now  be  conducted  by  sodium  light,  and  the  fluids  changed  until 
the  border  totally  disappears.  For  anisotropic  crystals  the  polarizer  is 
used  to  transmit  the  light  in  a  single  plane. 

226.  Marpmann  (1896). — For  embedding  diatoms  and  other  biological 

1  Compare  Sorby's  work,  Art.  212  and  Pauly's,  Art.  232. 

2  H.  Ambronn:      Farbenerscheinungen   an  den  Grenzen  farbloser   Objecte    im   Mikro- 
skop.     Ber.  Gesell.  Wiss.  Leipzig  ,  Math.-phys.  KL,  XL VIII  (1896),  134-140. 

3  Compare  the  explanation  given  by  Schroeder  van  der  Kolk,  Art.  228. 


ART.  227] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


255 


specimens,  so  that  their  structures  would  stand  out  in  relief,  Marpmann,1 
in  1896,  used  cinnamon,  cassia,  and  other  oils. 

227.  Schroeder  van  der  Kolk  (1898).— Schroeder  van  der  Kolk,2  in 
1898,  made  use  of  the  immersion  method  for  determining  refractive  indices, 
but  he  had  not  yet  discovered  its  quantitative  possibilities.  He  depended 
upon  the  width  and  strength  of  the  dark  border  to  determine  the  difference 
in  the  indices — the  wider  the  border,  th.e  greater  the  difference — and  called 
it  a  rapid,  even  though  not  very  exact,  method.  He  suggested  the  use 
of  a  series  of  fluids  of  known  indices  as  indicators  and,  since  certain  fluids 
act  as  solvents  for  certain  salts,  he  gave  two  lists  of  immersion  liquids,  one 
for  inorganic  substances,  and  one  for  organic.  Each  series  is  composed  of 
liquids  which,  in  most  cases,  can  be  mixed  with  each  other  in  any  quantity, 
thus  giving  the  possibility  of  preparing  fluids  of  any  desired  index. 


For  inorganic  salts 

n 

Sp.gr. 

Boiling 
point 

For  organic  salts 

n 

Sp.gr. 

Boiling 
point 

Hexane 

-37 
•39 
.46 
-47 
•49 
•50 
-50 
•5° 
•5i 
-53 
.56 
.60 
-63 

.66 
-76 
-95 

0.66 
0.71 
0.92 
0.92 
0.96 
0.89 
0.86 
0.92 
0.98 

1-05 
0.99 
i  .04 
1.29 

1.50 
3-34 

I  .  12 

68° 
98° 
174° 

Methyl  alcohol 

•32 
•34 
-36 
•37 
.40 

•45 
-47 
•54 
.60 

1.70 
1.72 
1.79 

2.  2O 

0.81 

I.  00 

0.72 
0.81 
0.83 
i-5o 
1.26 
i.  06 
1.04 

3-6o 
3-20 
3-59 

66° 

100° 

? 

132° 
61° 
290° 

2CO°  + 

183° 

Heptane  
Cajeput  oil  
Olive  oil 

Water 

Ethyl  ether 

Ethyl  alcohol  
Arayl  alcohol  
Chloroform  

Castor  oil  

265° 
80° 
136° 

"237°' 
253° 

220° 

180° 
47° 

277° 
180° 
272° 

Benzol  

Xylol  

Glycerine  

Beech  nut  oil  
Cedar  oil 

Creosote 

Aniline 

Clove  oil 

Cadmium  borotung- 
state  

Anise  oil  

Bitter  almond  oil  .  .  . 
Carbon  bisulphide.  . 
«   Monobromnaph- 
thalene 

Potassium  mercuric 
iodide 

Barium        mercuric 
iodide 

Methylene  iodide  .  .  . 
Phenyl  sulphide  

Mercuric   iodide   in 
aniline  and  quiniline. 

Intermediate  fluids  may  be  prepared  by  the  mixture  of  two  according  to  the 
formula 


n, 

where  v\  and  v%  are  their  respective  volumes.  Thus  9  volumes  of  heptane 
and  2  of  benzol  give  a  fluid  having  a  refractive  index  close  to  1.41.  Only 
fluids  having  approximately  the  same  boiling-points  should  be  combined, 
otherwise,  on  account  of  the  evaporation  of  one  component,  the  index  of 
the  mixture  may  change  rapidly.  He  suggests  mixtures  of  olive  and  castor, 

1  G.  Marpmann:  Ueber  die  Anwendung  von  Zimmtol,  Cassiaol,  und  anderen  Ein- 
schlussmitteln  in  der  Mikroskopie.  Zeitschr.  f.  angew.  Mikrosk.,  II  (1896-7),  335-338. 

-J.  L.  C.  Schroeder  van  der  Kolk:  Kurze  Anleitung  zur  mikroskopischen  Krystall- 
bestimmung.  Wiesbaden,  1898,  11-^14. 


256  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  228 

clove  and  cedar,  clove  and  bitter  almond,  and  anise  and  bitter-almond  oils, 
and  a-monobromnaphthalene  and  bitter  almond  oil.  Mixtures  of  clove  and 
anise  oil  become  cloudy  and  should,  consequently,  not  be  used.  Carbon 
bisulphide,  being  highly  volatile,  should  not  be  mixed  with  other  components. 
He  further  states  that  phenyl  sulphide  appears  not  always  to  have  the  same 
index  and  that  mercuric  iodide  in  aniline  and  quiniline  were  not  personally 
tried  by  him. 

To  determine  the  index  of  refraction  of  a  mineral  he  worked,  in  the  begin- 
ning, with  the  condenser  inserted.  The  process  is,  under  this  condition, 
less  sensitive,  and  the  borders  disappear  with  greater  differences  between 
the  indices  of  the  solid  and  the  liquid.  When  this  had  taken  place,  the  con- 
denser was  removed  and  the  limits  determined  with  greater  accuracy.  For 
still  greater  accuracy,  a  small  diaphragm  was  inserted,  and  finally  monochro- 
matic light  was  used. 

Van  der  Kolk1  speaks  of  colored  borders,  due  to  dispersion,  appearing 
when  the  black  border  disappears,  but  makes  no  further  use  of  them. 

228.  Schroeder  van  der  Kolk  (1900). — No  great  use  was  made  of  the 
immersion  method  until  it  received  its  great  impetus  by  the  publication  of 
Schroeder  (/an  der  Kolk's  Tabellen.2  After  the  issue  of  his  Anleitung,  which 
was  intended  primarily  for  chemists,  he  greatly  developed  the  method,  and 
it  was  here  made  use  of  for  the  rapid  determination  of  minerals,  some  300 
being  given  in  the  order  of  their  indices. 

According  to  former  methods,  the  dark  borders  enabled  one  to  determine 
that  solid  and  immersion  fluid  were  of  different  refractive  indices,  yet  one 
might  be  uncertain  whether  the  index  of  the  fluid  was  too  low  or  too  high. 
The  method  here  described  is  based  on  the  principle  of  the  dispersion  of  light 
by  prisms,  since  the  grains  of  crushed  minerals  have,  in  general,  more  or  less 
wedge-shaped  edges. 

If  the  condenser  and  polarizer  are  removed  from  the  microscope  and  a 
beam  of  monochromatic  light  is  directed,  by  means  of  the  plane  mirror, 
squarely  upon  a  more  or  less  lens-shaped  mineral  fragment  embedded  in  a 
liquid  of  a  different  refractive  index,  one  of  two  things  will  take  place.  If 
the  immersion  fluid  has  a  refractive  index  which  is  lower  than  that  of  the 
mineral,  the  latter  will  act  as  a  double  convex  lens  and  the  rays  will  first 
converge,  then  cross,  and  finally  diverge  (Fig.  356).  If,  on  the  other  hand, 
the  refractive  index  of  the  mineral  is  less,  it  will  act  as  a  double  concave  lens, 
and  the  rays,  after  passing  through,  will  diverge  (Fig.  357)  If  the  objective 
is  one  of  rather  low  power  and  has  a  considerable  focal  length,  the  appear- 
ance is  the  same  in  either  case  since. the  border  rays  will  be  deflected  too 
much  to  enter  the  lens,  consequently  a  dark  border  will  appear  around  the 

1  Op.  cit.,  page  45. 

2  J.  L.    C.    Schroeder   van  der   Kolk :  Tabellen  zur  mikroskopischen  Bestimmung  der 
Miner  alien  nach  ihrem  Brechungsindex.     2  Aufl.     Wiesbaden,  1906. 


ART.  228]  OBSERVATIONS  BY  ORDINARY  LIGHT  257 

mineral.1  So  far  as  the  border  is  concerned,  one  can  determine  only  that 
the  refractive  indices  of  the  two  media  are  different.  If,  now,  the  mirror  is 
swung  to  one  side  so  that  the  illumination  is  inclined  (in  Figs.  358  and  359 
from  the  right),  the  light  will  converge  or  diverge  as  before,  but  the  appear- 
ances as  seen  under  the  microscope  are  different.  In  the  case  where  the 
index  of  the  mineral  is  greater  than  that  of  the  liquid  (Fig.  358),  the  rays 
will  cross  and  diverge,  it  is  true,  but  the  one  on  the  opposite  side  from  that 
from  which  the  light  proceeded  will  more  or  less  directly  enter  the  lens,  and 
that  side  of  the  mineral,  or  the  opposite  side  of  the  image  seen  in  the  micro- 
scope, will  appear  bright.  When  the  refractive  index  of  the  mineral  is  lower 
than  that  of  the  liquid,  the  reverse  phenomenon  will  take  place  (Fig.  359). 


FIG.  356.  PIG.  357.  FIG.  358.  FIG.  359. 

FIGS.  356  TO  359. — The  cause  of  dark  or  light  borders. 

Instead  of  displacing  the  mirror,  inclined  illumination  may  be  produced 
much  more  readily  by  inserting  the  condenser  and  placing  above  it,  but 
below  the  section,  an  opaque  screen  of  thin  metal  or  of  cardboard,  extending 
to  a  greater  or  less  distance  beyond  the  middle. 

From  these  phenomena  is  derived  the  following  rule  for  observations 
made  without  the  condenser,  the  apparent  position  of  the  shadow  due  to  the 
inversion  of  the  image  by  the  microscope  being  taken  into  account.  When 
the  dark  shadow  appears,  in  the  image,  on  the  same  side  as  that  from  which  the 
screen  was  actually  inserted,  the  index  of  refraction  of  the  mineral  is  greater 
than  that  of  the  immersion  fluid;  when  it  appears  on  the  opposite  side,  the  refrac- 
tive index  of  the  mineral  is  less.  When  the  condenser  is  inserted  and  some- 
what lowered,  the  phenomenon  is  reversed.2 

Instead  of  using  a  screen  between  condenser  and  object,  inclined  illumi- 
nation3 may  be  produced  by  simply  shutting  off  part  of  the  light  by  holding 
the  finger  between  the  mirror  and  the  condenser,  by  a  sliding  diaphragm  such 

1  Compare,  here,  the  Becke  method,  Art.  236. 

2  This  method  of  illumination  and  the  phenomena  observed  had  already  been  described 
by  Becke.     See  Art.  239,  infra. 

3  J.  L.  C.  Schroeder  van  der  Kolk:     Uebcr  die   Vortheilc  schiefer  Beleuchtung  bel  der 
Unter  suchung  ion  Diinnschliffen  im  parallelen  polarisirten  Lichte.     Zeitschr.  f.  wiss.  Mik- 
rosk.,  VIII  (1891),  458. 

17 


258  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  228 

as  are  shown  in  Figs.  251,  252,  and  253,  or  by  a  sliding  diaphragm  in  the 
Ramsden  disk  above  the  ocular.1 

From  the  above  method  it  is  easy  to  determine  whether  the  refractive 
index  of  the  mineral  is  higher  or'lower  than  that  of  the  immersion  fluid.  To 
determine  how  much  they  differ,  use  is  made  of  the  color  effect  produced  by 
the  difference  in  the  dispersion  of  white  light  in  the  two  substances.  In 
general  this  is  greater  in  liquids  than  in  solids,  a  phenomenon  to  which  atten- 
tion had  already  been  called  by  Ambronn.2  If  the  refractive  indices  of  the 
two  substances  are  nearly  the  same  for  yellow  light,  the  solid  will  have  a 
higher  index  for  reds  and  a  lower  for  blue,  consequently  the  edge  of  the 
mineral,  in  the  image,  will  appear  blue  on  the  same  side  as  that  from  which 
the  screen  was  inserted,  and  orange  on  the  opposite  edge. 

Isotropic  Substances. — Since  isotropic  substances  have  but  a  single  refrac- 
tive index,  the  method  described  above  may  be  used  by  simply  immersing  a 
fragment  of  the  mineral  successively  in  fluids  of  different  indices.  The 
mineral  should  be  crushed,  not  powdered;  the  size  of  the  grains  being  such 
that  they  can  be  totally  submerged  in  a  drop  of  the  liquid  placed  upon  an 
object-glass.  The  minimum  size  that  may  be  used  is  such  that  the  two 
boundaries  of  the  grain  may  still  be  distinguished  with  an  objective  whose 

magnification  (T,J  is  about  15.     Begin  with  a  fluid  of  intermediate  index, 

and,  after  having  determined  whether  the  index  of  the  mineral  is  above  or 
below  that  of  the  fluid,  notice  whether  the  borders  are  heavy  and  black,  or 
whether  they  are  colored.  If  the  former  is  the  case,  the  indices  of  the  two 
substances  are  far  apart,  if  colored,  close  together.  Choose,  now,  another 
immersion  fluid  of  a  lower  index  if  the  one  first  tried  was  too  high,  or  of  a 
higher  index  if  it  was  too  low.  If  the  index  of  the  second  fluid  is  still  on  the 
same  side,  choose  a  third,  and  so  on.  If  it  falls  on  the  opposite  side,  work 
between  the  values  of  the  last  two  trials.  When  the  color  phenomenon  is 
produced,  use  monochromatic  light  and  change  the  immersion  fluid  until 
the  dark  boundaries  of  the  mineral  totally  disappear.  (Compare  the  Becke 
method,  Chapter  XV.) 

Anisotropic  Minerals.     Uniaxial  Crystals. — For  some  purposes,  the  mean 

value  of  the  refractive  indices  of  an  anisotropic  mineral  ( or ; j 

may  be  sufficient.  If  it  is  required  to  obtain  the  exact  values,  the  deter- 
mination is  more  difficult.  Both  analyzer  and  polarizer  should  first 

1  For  a  further  discussion  of  inclined  illumination  see 

H.  Schneiderhohn:  Die  Beobachtung  der  Interferenzfarben  schiefer  Strahlenbiindel  als 
diagnostisches  Hilfsmittel  bei  mikroskopischen  Miner aluntersuchungen.  Zeitschr.  f.  Kryst, 
L  (1912),  231-241. 

F.  E.  Wright:  Oblique  illumination  in  petrographic  microscope  work.  Amer.  Jour. 
Sci.,  XXXV  (1913),  63-82. 

2  Art.  226,  supra. 


ART.  229]  OBSERVATIONS  BY  ORDINARY  LIGHT  259 

be  inserted  and  the  stage  rotated  until  the  mineral  is  at  extinction.  In  this 
position  its  vibration  directions  correspond  with  those  of  the  nicols.  The 
analyzer  should  now  be  removed  and  the  refractive  indices  be  determined  in 
these  two  directions. 

Of  the  two  values  determined,  one  will  be  that  of  the  ordinary  ray.  It 
may  be  recognized  by  being  the  same  in  every  grain,  whatever  may  be  the 
orientation,  and  its  value  is  the  true  value  of  co.  In  grains  in  which  there  is 
obtained  an  interference  figure1  showing  the  emergence  of  the  optic  axis 
at  the  center,  the  values  will  be  the  same  regardless  of  the  azimuth  to  which 
the  stage  is  rotated. 

The  value  of  the  other  refractive  index  may,  or  may  not,  be  the  one 
desired,  since  a  section  through  the  index  surface  of  the  extraordinary  ray  is 
an  ellipse,  and  the  real  value  of  the  extraordinary  index  is  along  its  maximum 
or  minimum  vibration  axis,  depending  upon  whether  the  mineral  is  negative 
or  positive.  To  obtain  the  accurate  value  of  c,  one  should  determine  the 
maximum  or  minimum  value  of  a  large  number  of  grains.  If  it  is  known 
that  certain  fragments  are  elongated  in  the  direction  of  crystallographic  c, 
the  value  of  e  may  be  determined  at  once. 

Biaxial  Crystals. — The  index  surface  of  a  biaxial  crystal  is  an  ellipsoid 
of  three  axes,  consequently  there  are  three  indices  to  be  determined.  The 
process  is  very  similar  to  that  just  given.  Determinations  are  made  on 
a  large  number  of  grains,  and  the  highest  and  lowest  values,  assumedly  the 
maximum  and  minimum,  are  taken  for  7  and  a.  The  value  of  0  may  be 
computed  by  the  formula 

cf(j*-F) 

tan  F  =720?2-«2)' 

when  the  size  of  the  optic  angle  can  be  measured.     Under  the  microscope 
(3  may  be  determined,  in  a  mineral  fragment  which  shows  the  emergence  of 
a  bisectrix  at  the  center  of  the  field,  by  making  the  measurement  in  a  direction 
at  right  angles  to  the  plane  of  the  optic  axes. 
There  is  no  ordinary  ray  in  biaxial  crystals. 

229.  Immersion  Fluids. — A  great  many  different  substances  have  been 
proposed  for  immersion  fluids.  Not  only  simple  substances  may  be  used, 
but  mixtures  of  several  as  well,  and  it  is  thus  possible  to  prepare  a  series, 
differing  by  any  desired  amount.  It  is  usually  necessary  that  the  liquids 
chosen  should  retain  constant  indices  during  the  process  of  measurement, 
and  only  in  special  cases  is  a  changing  index  permissible.  The  boiling-points 
of  two  liquids  which  are  to  be  mixed  should  be  approximately  the  same,  other- 
wise one  will  evaporate  more  rapidly  than  the  other,  and  the  refractive  index 
will  vary.  The  liquids,  also,  should  be  unaffected  by  air,  so  that  the  indices 
of  the  stock  material  will  remain  constant  and  will  not  require  testing  every 

1  Chapter  XXIX,  infra. 


260 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  229 


time  that  they  are  used.  Neither  should  the  liquids  act  upon  the  minerals 
under  examination,  the  lens,  or  the  lens  casing.  Fulfilling  these  require- 
ments, the  various  oils,  in  particular,  are  well  adapted  for  immersion  fluids. 

In  the  following  table  the  values  given  are,  in  general,  those  at  20°  C. 
(68°  F.),  or  room  temperature.  In  some  of  the  fluids  there  is  a  decided 
variation  with  temperature,  and  if  the  work  is  performed  in  a  very  cold  or  a 
very  warm  room,  it  may  be  necessary  to  check  the  values  of  the  stock  material. 
A  greater  variation,  however,  than  that  produced  by  heat,  is  to  be  found  in 
different  lots  of  the  same  material,  even  of  the  same  make,  and  it  is  neces- 
sary, consequently,  to  test  the  values  of  each  new  purchase.  With  unques- 
tionably accurate  measurements,  different  determinations  have  given,  with 
some  fluids,  results  varying  as  much  as  0.04,  and  while  the  values  in  the  list 
below  were  probably  accurate  for  the  material  tested,  it  is  not  a  safe  pro- 
cedure to  accept  them  for  the  proper  indices  in  preparing  a  set  of  immer- 
sion fluids. 

In  some  of  the  older  text-books,  materials  of  very  high  indices  are  ap- 
parently incorrectly  given.  Thus  phenyl  sulphide  has,  according  to  Zirkel1 
and  Behrens,2  a  value  of  I.Q5,3  while  Himmelbauer4  found  it  to  be  1.638 
at  18.5°  C.  by  sodium  light.  Mercury  iodide  in  aniline  and  quiniline  is 
given  by  Zirkel  as  2.2,  but  Schroeder  van  der  Kolk,5  in  spite  of  repeated  at- 
tempts to  attain  a  refractive  index  so  high,  was  unable  to  succeed.  Wright6 
reached  a  value  of  only  1.8. 

TABLE  OF  REFRACTIVE  INDICES  OF  VARIOUS  IMMERSION  LIQUIDS 
(Arranged  in  the  order  of  increasing  values) 


Substance 

Index 

Temp. 

M 
3 

Sp.  gr. 

Boil.- 
pt. 

Formula 

Authority 

Air  ..  

ooooo 

Water.  
Water  
Ethyl  ether  
Acetone  
Ethyl  alcohol  
Ethyl  alcohol  .  .  . 

•33358 
•33392 
•35210 
•35932 
.36164 
.  36138 

i8.75 
15-25 
21.3° 

20.0° 
20.0° 
20   0° 

D 
D 
D 
D 
D 
D 

I  .00 

0.71 
0.82 
0.79 

100° 

"$' 

78° 

H2O 

'c4HioO"  ' 
C3H60 
C2H60 

Fraunhofer. 
Bailie. 
Lorenz. 
Korten. 
Ketteler. 
Korten. 

Hexane  
Heptane  
Chloroform  

•37536 
.3867 
.44366 

20.0° 

23.o: 
20.  o 

D 
D 
D 

0.66 
0.68 
1.49 

55°  ± 
98° 
61° 

C6H14 
CyHie 
CHC13 

Briihl. 
Gladstone. 
Lorenz. 

1  F.  Zirkel:  Lehrbuch  der  Petro graphic.     I,  2te  Aufl.,  Leipzig,  1893,  40. 

2  Wm.  Behrens:  Tabellen  zum  Gebrauch  bei  mikroskopischen  Arbeiten.     Braunschweig, 
3te  Aufl.,  1898,  50. 

3  Perhaps  the  9  in  the  original    reference  was,  by  a  typographical  error,  an  inverted  6. 

4  Alfred  Himmelbauer:    Bemerkungen  tiber  das  Phenylsulfid.      Centralbl.  f.  Min.,  etc., 
1909,  396. 

6  Schroeder  van  der  Kolk:  Tabellen,  etc.,  p.  n. 

6  Fred  Eugene  Wright:    The  methods  of  petrographic-microscopic  research.     Carnegie 
Publication  No.  158,  Washington,  1911,  98,  footnote. 


ART.  229] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


261 


TABLE  OF  REFRACTIVE  INDICES  OF  VARIOUS  IMMERSION  LlQUlDS.-Conlinued 


Substance 

Index 

Temp. 

+J 

J3 

be 

3 

Sp.  gr. 

Boil.- 
pt. 

Formula 

Authority 

Ethylene  chloride  .... 
Petroleum  

44439 

4.C 

20.0° 

D 

1-25 

84° 

C2H4C12 

Weegmann. 
Wright. 

Lavender  oil!  
Carbon  tetrachloride  .  .  . 
Turpentine1    
Glycerine   

.461 
.4656 
.47212 
472Q3 

"i2".30' 
20.7° 
20  o° 

b 

D 

D 

0.88 
1.61 
0.89 
i   26 

188° 

,£ 

ecu 

CioHie 
C3H8O3 

S.  v.  d.  Kolk. 
Gladstone, 
v.  d.  Willigen. 
Landolt 

Olive  oil...  
Beechnut  oil  
Almond  oil  

•  4763 

•477 
.4782 

0.0° 

00° 

D 

b 

0.92 
0.92 

Olds. 
S.  v.  d.  Kolk. 
Olds. 

Castor  oil 

478 

o  06 

265°  + 

S  v.  d.  Kolk 

Castor  oil  . 

481 

16  o° 

Behrens. 

Toluol  
Xylol,  ortho-  
Xylol,  meta-  
Xylol    para-  . 

•49552 
.4966 
.5020 
4.84.6 

20.0° 

18.0° 

15.5° 

16  o° 

D 
D 
Dr 
D 

0.87 
0.86 
0.87 
o  8s 

110° 

136° 

C7H8 
CgHio 
C8H10 
C8Hio 

Briihl. 
Gladstone. 
Gladstone. 
Gladstone 

Benzol2   
Pseudocumol  
Sandal  wood  oil  

•4979 
.4801 
^07 

21-5° 
12.0° 

16  o° 

D 
D 

0.88 
0.84 

80° 
170° 

C6H6 
C-9H12 

Gladstone. 
Gladstone. 
Behrens. 

Ethyl  iodide  

.  51307 

20  o° 

P 

I    03 

73° 

C2H5I.  ..  . 

Haagen. 

Cedar  wood  oil1  
Cedar  wood  oil  
Monochlor  benzol  

•5i6 
.510 

•  527 

16.0° 

0.98 

I  .  13 

& 

1^2° 

S.  v.  d.  Kolk. 
Behrens. 
S.  v.  d.  Kolk. 

Ethylene  bromide3  .... 
Fennel  oil  .  . 

.53789 
538 

20.0° 

D 

2.18 

o  06 

130° 
188° 

C2H4Br2 

Weegmann. 
S.  v.  d.  Kolk. 

Canada  balsam 

54± 

See  Art.  243. 

Clove  oil1    

CAA 

I    O5 

253° 

S.  v.  d.  Kolk. 

Clove  oil  

.  C72 

Behrens 

Bitter  almond  oil  
.Anise  oil  
\itro  benzol* 

•54638 

•54754 

r  C2OI 

20.0° 

21-4° 
20    0° 

D 
D 
T) 

1.05 
0.98 
i  20 

220°" 

209° 

C7H60 
CeHsNOa 

Landolt. 
v.  d.  Willigen. 
Briihl 

Dimethyl  aniline  
Afonobrombenzol  

.55873 

561 

20.0° 

D 

0.96 

I     52 

o 
I92° 

ICC° 

CsHnN 

Briihl. 
S.  v.  d.  Kolk. 

Orthotoluidine4  

572 

O   09 

108° 

S.  v.  d.  Kolk. 

Aniline1'    4>  6    
Bromoform*'  5    
Monochloranaline  .... 

.58629 
.5890 
592 

20.0° 

20.0° 

D 
D 

I  .02 
2.82 

•  24 

I380 
151° 

207° 

C6H7N 
CHBr3 

.  Briihl. 
Jahn. 
S.  v.  d.  Kolk. 

Cassia  oil  

6026 

2?    =5° 

D 

O4, 

Baden-Powell. 

Cassia  oil  

58624 

20  o° 

F> 

Wiedemann. 

Quiniline1'    7  
Quiniline1'  *  
Cinnamon  oil1'  6    .... 
M  onoiodbenzol  
Carbon  bisulphide8  .  .  . 
Phenyl  sulphide  

.6171 
.6262 
.61879 
.621 
.62761 
635 

20.  C° 
IO.O° 

23.5° 


20.0° 

18  5° 

D 
D 

D 

b 

D 

.09 
.  IO 
.06 

-83 
.26 

237° 

225°" 
188° 
46° 

C9H7N 
C9H7N 

"cs2" 

Berliner. 
Gladstone, 
v.  d.  Willigen. 
S.  v.  d.  Kolk. 
Ketteler. 
Himmelbauer. 

a-Monochlornaphtha- 
lene 

•639 

1.20 

260° 

a-Monobromnaphtha- 
lene. 
ce-MonobromnaphtJia- 

i  .  64948 
i  65114 

20.0° 
16   s° 

D 

I    50 

277° 

Walter. 
Xasini. 

Icne. 
ot-Monobromnaphtha- 

i  66102 

23   5° 

Dufet 

lene. 
Phosphorus   tribrom- 

i  6866 

25    0° 

Zirkel. 

ide. 
Cadmium    borotung- 

i   70 

Zirkel. 

state  solution.9 
Potassium     mercuric 

i   7167 

18  o° 

D 

3    II 

Goldschmidt. 

iodide  solution.10 

See  end  of  table  for  notes. 


262 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  229 


TABLE  OF  REFRACTIVE  INDICES  OF  VARIOUS  IMMERSION  LIQUIDS.— Continued 


Substance 

Index 

Temp. 

«j 

jc; 

bC 

'3 

Sp.gr. 

Boil.- 

pt. 

Formula 

Authority 

Methylene  iodide11  .... 
Methylene  iodide11  .... 
Barium  mercuric  io- 
dide solution.12 
Sulphur  in  methylene 

1.7421 
1-7559 
i-793i 

i  8 

19.0° 

I0< 
23.0° 

D 

s 

3-32 
3-34 
3-564 

1.81° 

CH2I2 
CH2I2 

Gladstone. 
Gladstone. 
Rohrbach. 

Bertrand 

iodide  

i  83 

S   v  d   Kolk 

Sulphur  in  methylene 
iodide  
Molten  sulphur  
Molten  sulphur  
Mercury  methyl  

i-79 
i.8g± 

i-93± 

I  .03 

'  130°  ' 

110° 

Wright. 
S.  v.  d.  Kolk. 
S.  v.  d.  Kolk. 
Zirkel. 

Phosphorus  in  €82...  . 

I  .9"5 

20.0° 

Retgers. 

Molten  phosphorus.  .  . 
Molten  phosphorus..  . 
Selenium  

2.075 
2.II3-II 

2  .92 

44-0° 
44.0° 

D 
H/? 

D 

i-75 

P 

Se 

Retgers. 
Damien. 
Merwin    & 
Larsen. 

1  Oxidizable.     Should  be  kept  from  air. 

2  Very  useful  for  cleaning  oil  from  minerals. 

3  Extremely  poisonous. 

4  Sensitive  to  light. 

6  If  the  grains  float,  use  a  cover-glass. 

6  Strong  dispersion. 

7  Hygroscopic.     Add  a  piece  of  KOH  to  the  liquid. 

8  Very  volatile  and  can  be  used  only  with  a  cover-glass.      It  should  be  allowed  to  flow 
under  the  edge  after  the  cover-glass  has  been  placed  over  the  mineral. 

9  Klein's  solution. 

10  Thoulet's  solution.     Very  poisonous. 

11  Sensitive  to  light.     The  iodine  which  separates  may  be  removed  with  copper. 

12  Rohrbach's  solution. 

In  the  above  table  the  names  printed  in  italics  are  those  recommended 
by  Schroeder  van  der  Kolk,  and  most  of  them  are  miscible. 

For  practical  use  in  petrographic  work,  the  difference  between  the  indices 
of  each  fluid  and  the  one  next  succeeding  it  need  not  be  less  than  0.005.  They 
should  be  kept  in  well-stoppered  bottles,  systematically  arranged  in  a  wooden 
rack  or,  better,  in  a  covered  box.  The  bottles,  doubly  closed  by  stopper  and 
cap,  and  provided  with  a  convenient  glass  dropper,  should  be  small  enough 
so  that  no  great  amount  of  fluid  is  necessary  to  fill  them,  the  half-ounce 
(15  c.c.)  size  being  ample.  Kept  in  such  bottles,  the  amount  of  change  in 
values  is  not  great.  A  set  of  oils,  prepared  by  the  writer  and  tested  after 
two  years,  showed  a  maximum  change  of  0.003.  £>e  Lorenzo  and  Riva1 
determined  the  indices  of  a  set  of  oils  after  three  to  six  months.  No  state- 
ment is  made  in  regard  to  the  care  taken  of  the  liquids  in  the  meantime. 
The  following  values  were  found,  tests  being  made  with  an  Abbe-Pulfrich 
refractometer. 

1  De  Lorenzo  and  Riva:  Review  in  Zeitschr.  f.  Kryst.,  XXXV  (1902),  501-502.  Die 
Krater  von  Vivara  aufden  Phlegre'ischen  Inseln,  Mem.  Roy.  Ace.  Sci.,  Napoli,  X  (1901),  1-60. 


ART.  229] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


263 


I(i8°)  I 

I  (18°) 

.4650  : 

.4644 

.4738 

.4850 

•4855 

.5090 

•5095 

.5178 

.5208 

•5193 

.5270 

.5280 

•5347 

•5336 

•5363 

•5365 

•5396 

•5412 

•5562 

•5563 

•  ^90 

oov 
•5750 

•5751 

.5840 

•5830 

•6033 

.5980 

Lavender  oil 

Cedar  oil 

Juniper  oil 

Fennel  oil 

Mixture  of  lavender,  fennel,  and  cinnamon  oils 

Mixture  of  lavender,  clove,  and  cinnamon  oils 

Clove  oil 

Mixture  of  clove  and  cinnamon  oils 

Wintergreen  oil 

Almond  oil 

Anise  oil 

Mixture  of  clove  and  cinnamon  oils 

Mixture  of  clove  and  cinnamon  oils 

Cinnamon  oil  (Goa) 

Cinnamon  oil  (Ceylon) 

Among  the  various  combinations  of  liquids  which  may  be  used,  those 
proposed  by  Wright1  are  very  good.  He  prepared  a  set  of  immersion  fluids 
as  follows  and  found  a  change  of  not  over  0.002  in  a  year.  For  temperature, 
there  is  a  decrease  of  about  o.ooi  for  every  3°  C. 

Mixtures  of  petroleum  and  turpentine 450— i  .475 

Mixtures  of  turpentine  and  ethylene  bromide  or  clove  oil 480—1 .535 

Mixtures  of  clove  oil  and  a-monobromnaphthalene 540— i  .635 

Mixtures  of  a-monobromnaphthalene  and  a-monochlornaphthalene 640-1 . 655 

Mixtures  of  a-monochlornaphthalene  and  methylene  iodide 660— i .  740 

Sulphur  dissolved  in  methylene  iodide 74o-i .  790 

Methylene  iodide,  antimony  iodide,  arsenic  sulphide,  antimony  sulphide,  and 

sulphur i .  790-1 . 960 

For  minerals  having  very  high  refractive  indices,  Merwin  and  Larsen2 
used  molten  sulphur,  molten  selenium,  and  mixtures  of  the  two,  these  sub- 
stances being  miscible  in  all  proportions  when  in  a  molten  condition.  The 
mixtures  are  prepared  by  placing  the  required  weight  of  powdered  selenium 
in  a  3-in.  test-tube,  heating  it  until  the  mineral  is  thoroughly  fused,  and  allow- 
ing it  to  cool.  The  proper  amount  of  pure  flowers  of  sulphur  is  now  added, 
and  the  mixture  heated  just  enough  to  allow  thorough  mixing  with  a  glass  rod. 
As  the  material  cools  it  is  gathered  on  the  rod,  and  finally  cut  into  small 
fragments.  These  may  now  be  returned  to  the  tube,  which  should  be 
corked,  and  preserved  for  use.  One  or  two  grams  are  sufficient  to  examine 
a  hundred  minerals. 

To  determine  refractive  indices  with  this  preparation,  a  small  piece  of  it 
and  a  little  of  the  mineral,  finely  pulverized,  are  heated  together  on  an  object- 

1  Fred.  Eugene  Wright:  Op.  cit.,  96. 

2  H.  E.  Merwin  and  E.  S.  Larsen:    Mixtures  of  amorphous  sulphur  and  selenium  as 
immersion  media  for  the  determination  oj  high  refractive  indices  with  the  microscope.     Amer. 
Jour.  Sci.,  XXXIV  (1912),  42-47. 

Both  sulphur  and  selenium  had  long  previously  been  used  as  immersion  fluids  with 
high  refractive  indices.  Mixtures  of  selenium  and  sulphur  or  arsenic  were  used  by  Marp- 
mann.  [G.  M(arpmann):  Das  Selen  als  Einschlussmittel  fur  Diatomaceen.  Zeitschr.  f. 
angew.  Mikrosk.,  IV  (1898),  6-8.]  Marpmann  also  used  selenium  dissolved  in  selenium- 
ethyl  Se(C2H5)2. 


264 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  229 


and  under  a  cover-glass,  over  a  small  flame,  until  the  preparation  is  liquid, 
when  the  two  are  mixed  and  pressed  into  a  thin  film.  The  film  is  again 
heated  for  half  a  minute  until  bubbles  begin  to  appear  when  it  is  again  pressed 
thin  and  cooled,  after  which  the  determination  is  made  in  the  usual  manner. 
With  care  no  appreciable  amount  of  sulphur  will  be  vaporized.  The  cooled 
mixtures,  rich  in  selenium,  have  a  deep  red  color  and  remain  amorphous  for 
months,  those  very  rich  in  sulphur  are  yellow  to  orange,  and  may  crystallize 
immediately  on  cooling.  With  less  than  15  per  cent.  Se  this  crystalliza- 
tion takes  place  so  readily  that  they  are  not  well  adapted  to  accurate  work. 

Owing  to  the  high  dispersion  of  the  selenium,  it  is  desirable  to  use  mono- 
chromatic light  for  accurate  work,  a  simple  method  being  to  make  a  screen 
by  pressing  a  bit  of  heated  selenium  between  an  object-  and  cover-glass,  and 
placing  it  on  the  eyepiece.  The  transmitted  light  gives  a  wave  length 
approximately  equivalent  to  that  of  lithium.  With  white  light  and  colorless 
minerals,  the  light  and  shade  effect  may  not  be  seen  in  large  grains  owing  to 
the  excess  of  illumination.  In  such  cases  the  smaller  grains,  which  are  more 
deeply  covered  by  the  mixture,  may  be  used. 

The  following  table  gives  the  refractive  indices  of  different  mixtures  for 
lithium  and  sodium  flames. 


Per  cent.  Se 

nLi 

nNa 

Equivalent  wave 
length  in  MM 

o.o 

1.978 

1.998 

9.0 

2  .OOO 

2  .022 

17.6 

2  .025 

2  .O5O 

25.0 

2  .050 

2.078 

31-8 

2.075 

2  .  IO7 

37-5 

2.100 

2.134 

43-2 

2.125 

2.163 

"580" 

48.2 

2  .  150 

2.193 

605 

53-o 

2-175 

2  .  22O 

615 

S7-o 

2  .  2OO 

2.248 

620 

64.0 

2.250 

2.307 

630 

70.0 

2.300 

2.365 

633 

75-o 

2-350 

2.423 

636 

80.0 

2  .400 

2.490 

640 

87-7 

2.500 

2  .624 

645 

93-8 

2  .6OO 

2-755 

652 

99-2 

2.700 

2  .90 

662 

IOO.O 

2  .  716 

2.92 

665 

To  fill  the  gap  between  fluids  having  refractive  indices  from  1.33  to  1.80 
and  from  2.1  to  2.4,  Merwin1  proposed  solutions  of  iodoform,  tri-iodide  of 
arsenic,  tri-iodide  of  antimony,  tetra-iodide  of  tin,  and  sulphur  in  methylene 
iodide.  With  various  proportions  dissolved  in  100  parts  of  methylene  iodide, 

1  H.  E.  Merwin:  Media  of  high  refraction  for  refractive  index  determinations  with  the 
microscope;  also  a  set  of  permanent  standard  media  of  lower  refraction.  Jour.  Washington 
Acad.  Sci.,  Ill  (1913),  35-40. 


ART.  231]  OBSERVATIONS  BY  ORDINARY  LIGHT  265 

fluids  of  refractive  indices  between  1.764  and  1.868  were  obtained.  Fluids 
from  1.74  to  2.28  were  obtained  by  dissolving  arsenic  trisulphide  in  methylene 
iodide  near  its  boiling-point.  Merwin  also  prepared  resin-like  substances 
with  indices  between  1.68  and  2.10  by  dissolving  tri-iodides  of  arsenic  and 
antimony  in  piperine.  For  media  between  2.1  and  2.6  he  used  mixtures 
of  amorphous  sulphur  and  arsenic  trisulphide.  Other  media  were  mixtures 
of  piperine  and  rosin  for  indices  between  1.546  and  1.682,  and  mixtures  of 
rosin  and  camphor  for  1.510  to  1.546. 

DETERMINATION  OF  THE  REFRACTIVE  INDICES  OF  FLUIDS 

230.  Introductory. — In  the  previous  method  for  the  determination  of 
the  indices  of  refraction  of  minerals,  it  is  required  to  have  liquids  of  known 
indices.     The  determination  of  the  indices  of  these  liquids  may  be  made 
most  accurately  with  an  Abbe-Pulfrich  refractometer,  but  such  an  instru- 
ment is  not  always  available,  and  a  method  for  determining  them  by  means 
of  the  microscope  itself  is  a  great  convenience,  especially  for  checking  the 
values  after  the  liquids  have  been  kept  on  hand  for  a  number  of  years. 

231.  Smith's   Method    (1885). — As    long    ago    as  1813  Brewster1  de- 
termined the  refractive  indices  of  liquids  by  an  application  of  the  Due  de 
Chaulnes'  method,  and  a  similar  method  was  given  by  Becquerel  and  Cahors2 
in  1840.     Both  methods  require  the  use  of 

a  considerable  amount  of  the  fluid  whose 
refractive  index  is  to  be  determined,  and 
the  result  must  be  computed  mathemat- 
ically. A  method,  based  on  the  same 
principle,  but  requiring  only  a  small 

J  FIG.     360. — Smith  s   apparatus  for   deter- 

a mount    of  material    and    no   Calculations,  mining  the  refractive  index  of  a  fluid. 

was  devised  by  Smith3  in  1885. 

The  instrument,  by  means  of  which  the  refractive  indices  of  fluids  are 
measured,  consists  of  a  short  cylinder  (A,  Fig.  360)  which  is  inserted  at 
the  lower  end  of  the  tube  of  the  microscope,  just  above  the  objective  clutch. 
Sliding  in  this  cylinder  are  two  slips  of  crown  glass  a  and  b,  2  in.  long,  1/2 
in.  wide,  and  i/ioin.  thick,  and  having  a  refractive  index  as  nearly  as  possible 
the  same  as  that  of  the  cover-glass.  One  of  these  slips  b  has  a  polished  con- 
cave depression,  one-third  or  more  of  the  thickness  of  the  glass,  ground  in 
it  near  one  end. 

1  Art.  209,  supra. 

2  Art.  210,  supra. 

3  H.  L.  Smith:    Device  for  testing  refractive  index.     Amer.  Mon.  Microsc.  Jour.,  VI 
(1885),  181-182. 

Idem:  Device  for  testing  refractive  index  of  immersion  fluids.     Proc.  Amer.  Soc.  Microsc., 
8th  annual  meeting,  Cleveland,  VII  (1885),  83-85. 


266  MANUAL  OF  PETROGRAPHIC  METHODS          ,        [ART.  232 

The  method  of  determining  the  refractive  index  of  a  fluid  is  to  place  a  drop 
of  it  in  the  depression  of  the  lower  glass  slip,  place  above  it  the  other,  thus 
squeezing  a  thin  film  of  the  medium  between  the  two,  and  insert  it  in  the  slot 
of  the  adaptor.  With  the  slips  in  the  position  shown  in  the  figure  and  with 
a  i  in.  objective  inserted  in  proper  position  beneath  it,  it  will  be  found  that 
there  has  been  no  appreciable  change  in  the  focus  of  the  instrument.  The 
microscope  is  now  focussed  sharply  upon  some  clearly  defined  object,  then 
the  slips  are  pushed  in  until  the  liquid  lens  lies  directly  back  of  the  objective. 
If  the  medium  is  homogeneous  with  the  glass  of  the  slips,  there  will  be  no 
change  in  focus  or  definition,  and  no  chromatic  aberration.  Since  no  im- 
mersion oil  known  is  strictly  homogeneous  in  this  sense,  although  it  may 
have  the  same  refractive  index  as  the  glass,  the  focus  may  be  unchanged 
although  a  colored  rim  will  appear.  If  the  fluid  being  tested  is  of  a  different 
refractive  index  it  will  be  necessary  to  change  the  focus  of  the  microscope, 
the  amount  depending  upon  the  value  of  the  index.  Working  with  a  few 
fluids  of  known  refractive  indices,  one  may  mark,  upon  the  side  of  the  tube, 
the  positions  of  the  focus  for  different  values,  the  rack  and  pinion  adjustment 
being  used,  the  fine  adjustment  remaining  continually  the  same.  Thus  if 
the  cavity  is  filled  with  cinnamon  oil  we  get  a  certain  mark  for  a  value  of 
1.6;  using  the  same  object,  objective,  and  eyepiece,  we  get  another  of  1.33 
for  water,  and  still  others  for  cedar-oil,  glycerine,  clove-oil,  etc.1  With  these 
points  scratched  on  the  side  of  the  tube,  by  interpolation  the  intermediate 
values  may  be  easily  determined,  the  distance  between  1.3  and  1.6  being 
about  half  an  inch. 

232.  Pauly's  Method  (1905).  —  Pauly's2  method  for  determining  refrac- 
tive indices  under  the  microscope  is  extremely  simple,  and  he  claims  that  it 
is  correct  to  2  or  3  in  the  fourth  decimal  place.  It  is  based  on  a  modifica- 
tion of  Ambronn's3  method. 

The  indicatrix  of  uniaxial  crystals  is  an  ellipsoid  of  rotation,  and 

_  coe 

" 


is  the  equation  of  the  index  of  refraction  of  a  wave  whose  normal  makes  an 
angle  of  <p  with  the  c  axis.  Every  section  of  the  indicatrix  is  an  ellipse  and, 
in  a  plate  cut  parallel  to  the  principal  axes,  all  values  of  indices,  intermediate 
between  the  values  of  the  principal  indices,  will  lie  on  the  ellipse. 

If  a  plate  of  calcite,  cut  parallel  to  c,  is  placed  on  the  stage  of  the  micro- 

1See  table  Art.  229. 

2  Anton  Pauly:    Ueber  eine  einfache  Methode  zur  Bestimmung  der  Brechungsexponenten 
iion  Fliissigkeiten.     Zeitschr.  f.  wiss.  Mikrosk.,  XXII  (1905),  344-348. 

3  H.  Ambronn:     Ueber  eine  neue  Methode  zur  Bestimmung  der  Brechungsexponenien 
anisotroper  mikroskopischer  Objecte.     Ber.  Akad.  Wiss.  Leipzig,  1893,  316-318.      The  method 
is  described  in  Art.  224,  supra. 


ART.  232] 


OBSERVATIONS  BY  ORDIXARY  LIGHT 


267 


scope,  and  upon  it  is  placed  a  drop  of  the  fluid  whose  index  (between  co  =  i  .6585 
and  e=  1.4864  of  the  calcite)  is  to  be  determined,  and  it  is  then  covered  with  a 
cover-glass,  it  will  be  found,  when  the  polarizer  only  is  inserted  and  the  dia- 
phragm beneath  the  stage  is  partially  closed,  that  upon  rotating  the  stage  a 
certain  amount,  practically  all  of  the  inequalities  on  the  surface  of  the  cal- 
cite, as  well  as  the  border  of  the  drop* will  disappear.1  The  cause  of  this  is 
that  the  polarizer  permits  only  rays  vibrating  parallel  to  one  direction  to 
pass  through  the  calcite  plate,  and  the  rays  which  reach  the  eye  have  an  index 
of  refraction  equal  to  that  radius  of  the  calcite  index-ellipse  which  is  parallel 
to  the  vibration  direction  of  the  nicol.  When  the  index  of  refraction  of  the 
calcite  plate  exactly  equals  that  of  g|  ^  ^  ^  ^  o  ^  2  gs? 
the  fluid,  which  is  isotropic,  all  in- 
equalities between  the  two  disap- 
pear and  the  light  passes  through 
as  though  there  were  but  a  single 
medium.  The  angle  A  at  disap- 


Vibration  (Urection\/x6f  ^?~  nicol 


FIG.  361. — Diagram  illustrating 
Pauly's  method  for  determining  re- 
fractive indices. 


FIG.  362. — Diagram  for  determining  the  indices 
of  refraction  of  liquids  by  the  amount  of  rotation 
of  the  stage  necessary  to  produce  a  disappearance  of 
the  boundaries  when  placed  on  a  calcite  plate  (90°-^). 


pearance  is  read  from  the  stage  graduations,  the  stage  is  turned  until  the 
lines  again  disappear,  and  the  angle  B  is  read.  1/2  (A—B)  =  <p  or  90°  — <p, 
as  the  case  may  be  (Fig.  361).  The  value  of  the  index  may  be  computed 
from  the  formula,  or  it  may  be  determined  graphically  from  the  diagram 
(Fig.  362).  One  must  note  whether  the  angle  <p  or  90°  — <p  was  read,  which 
is  easily  done  by  making  a  mark  on  the  calcite  in  the  direction  of  the  c  axis 
(the  direction  of  c).  The  diagram  is  computed  for  angles  from  c  to  co. 

If  no  orientated  section  is  at  hand,  a  cleavage  plate  of  calcite  may  be 
taken,  but  here  the  range  in  indices  is  only  from  1.567  to  1.658.  For  fluids 
having  a  higher  index  than  o>  of  calcite,  siderite,  in  which  €=1.643  and  «  = 
1.872  may  be  substituted.  Pure  calcite  or  siderite  must  be  used,  otherwise 
the  refractive  indices  will  be  different  from  those  here  given. 

The  writer  has  been  unable,  by  Pauly's  method,  to  obtain  results  closer 
than  3  in  the  second  decimal  place. 

1  Cf.  the  method  of  Schroeder  van  der  Kolk,  Arts.  227-228. 


268  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  233 

233.  Michel-Levy's  Indicators. — Since  the  refractive  index  of  a  mineral 
may  be  determined  by  immersing  it  in  a  fluid  of  known  index,  inversely 
that  of  the  fluid  may  be  found  by  immersing  in  it  a  mineral  of  known  index. 

A  series  of  such  indicators  was  proposed  by  Michel-Levy,1  in  1894. 
Each  fragment  was  oriented  in  a  definite  direction  and  all  were  mounted  on  a 
number  of  glass  plates  in  the  order  of  increasing  indices.  The  scale  was 
composed  of  the  following  minerals: 

Fluorite 433 

Hauynite 496 

Leucite 508 

Orthoclase 526  and  1.519 

Microcline 529  and  i . 523 

Albite 540  and  i .  532 

Cordierite 589  and  i .  532 

Oligoclase 542  and  i .  543 

Nephelite 547  and  i .  543 

Quartz 553  and  i .  544 

Andesine 556  and  i .  549 

Labradorite 562  and  i .  554 

Anorthite 588  and  i .  575 

Melilite 641  and  1.621 

Apatite 638  and  i .  634 

Andalusite i .  643  and  i .  632 

The  objection  to  the  indicators  of  Michel-Levy  is  that  not  only  may  the 
refractive  indices  of  the  same  minerals  be  different  in  specimens  from  dif- 
ferent localities,  but  even  in  those  from  the  same  quarry,  consequently  each 
mineral  used  must  be  carefully  tested  before  being  mounted.  Another 
objection  is  that  it  is  difficult  to  find  minerals  differing  by  uniform  amounts, 
and  a  third,  that  anisotropic  crystals  with  different  indices  in  different 
directions  must  be  used  for  some  of  the  indicators. 

234.  De  Souza-Brandao's  Indicators. — The  indicators  proposed  by  de 
Souza-Brandao2  and  prepared  by  Fuess  are  made  of  small  squares  of  glass, 
2  mm.  on  a  side  and  i  mm.  thick,  and  of  different  refractive  indices.     Being 
isotropic,  the  values  are  the  same  in  all  directions,  and,  since  glass  is  amor- 
phous and  homogeneous,  many  squares  can  be  cut  from  one  specimen  whose 
index  of  refraction  is  accurately  determined,  a  great  advantage  in  the  com- 
mercial preparation  of  such  scales.    The  scale  consists  of  35  different  indi- 
cators mounted,  2.5  mm.  apart,  in  Canada  balsam,  on  seven  object  slips, 
47  mm.  X  27  mm.,  and  with  the  index  values  engraved  on  the  glass  opposite 
each.     The  seven  slides  are  as  follows: 

1  A.  Michel-L6vy:  Etude  sur  la  determination  des  feldspaths.  Premiere  fascicule, 
Paris,  1894,  62-63. 

2V.  de  Souza-Brandao:  Ueber  eine  Skala  von  Lichtbrechungs-Indicatoren.  Centralb. 
f.  Min.  etc.,  1904,  14-18. 


ART.  234] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


269 


II 


III 


IV 


VI 


VII 


•434 

1-494 

-523 

•552 

•590      1-631 

.680 

•450 

I  .501 

•531 

-558 

.  604      i  .  648 

•693 

•465 

1-509 

.536 

•564 

.614      1.657 

.702 

.478 

1.512 

•539 

•573 

.620      1.666 

.717 

.486 

1-516 

.548 

.580 

•625      1.673 

•735 

To  determine  the  index  of  an  unknown  liquid,  a  few  drops  are  placed  on 
the  indicator  and  covered  with  a  cover-glass,  and  the  relative  indices  noted, 
either  by  inclined  illumination  or  by  the  Becke  method  described  below. 
Another  method  is  to  fill  a  small  glass  tray  to  a  depth  of  1/2  mm.  with  the 
liquid  whose  index  is  to  be  determined  and  invert  in  it  one  of  the  test  plates. 
Such  a  tray,  1 1  mm.  deep  and  2  mm.  longer  and  broader  than  the  indicator 
slips,  is  furnished  with  each  scale.  The  determinations  of  the  relative  in- 
dices are  made  through  the  object  slip  and  are  accurate  to  about  0.003. 

De  Souza-Brandao  also  suggested  that  instead  of  using  so  many  dif- 
ferent refractive  oils,  one  would  better  use  a  single,  dilutable  fluid.  For 
this  purpose  Sonstadt's  (Thoulet's)  solution1  is  excellent.  It  is  miscible 
with  water  in  all  proportions,  and  does  not  act  upon  Canada  balsam,  the 
cement  of  the  scale.  The  action  upon  Canada  balsam,  after  a  time,  of  many 
of  the  oils  and  of  a-monobromnaphthalene  and  methylene  iodide,  is  a  great 
objection  to  their  use  with  these  indicators.  Methylene  iodide  is  not  well 
adapted  for  general  use  in  the  determination  of  refractive  indices,  since  the 
fluids  with  which  it  is  miscible  are  very  volatile,  and  the  refractive  index  of 
the  mixture  changes  rapidly.  Sonstadt's  solution  may  most  conveniently 
be  made  up  in  ten  different  strengths,  having  specific  gravities  of  1.5,  1.7, 
1.9,  2.1,  2.3,  2.7,  2.9,  3.0,  and  3.1,  and  corresponding  to  indices  ranging  from 
1.42  to  i-72.2  The  mixtures  should  be  kept  in  not  too  small  pipette  flasks. 
In  each  bottle  two  specific  gravity  indicators3  may  be  placed,  such  that 
one  just  floats  and  one  sinks  when  the  liquid  is  of  the  proper  specific  gravity, 
consequently  of  proper  index. 

The  advantage  of  using  but  one  kind  of  fluid  is  that  after  approximately 
determining  the  refractive  index  of  the  mineral  by  immersion  tests,  a  con- 
siderable quantity  of  the  stock  solution,  nearest  this  value,  may  be  slightly 
diluted  until  it  reaches  the  exact  index  of  the  mineral.  Its  exact  value  may 
now  be  determined  by  means  of  a  refractometer,  Pauly's  method,  or  the 
above  mentioned  scale.  The  used  material  may  then  be  evaporated  a  trifle 
on  the  water  bath4  until  it  reaches  a  specific  gravity  slightly  greater  than 
that  of  the  stock  material  from  which  it  was  taken.  After  pouring  back,  a 

1  Art.  454. 

2  Art.  215. 

3  Art.  481. 

4  Art.  454. 


270 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  235 


few  drops  of  water  will  bring  it  to  the  proper  specific  gravity.  The  objec- 
tion to  Sonstadt's  solution  is  that  its  first  cost  is  considerable  and  it  is  ex- 
tremely poisonous. 

235.  Clerici's  Method  (1907). — A  simple  method  of  directly  determining 
the  refractive  index  of  a  fluid,kunder  the  microscope,  was  proposed  by  Clerici. l 
It  has  the  advantage  of  requiring  no  change  of  indicators,  and  may  be  used 
to  determine  the  refractive  index  of  a  volatile  fluid  with  changing  index  the 
instant  it  corresponds  with  that  of  the  solid  immersed. 

The  apparatus  (Fig.  363)  consists,  simply, 
of  an  object  slip,  in  the  center  of  which  two 
lines  are  engraved  crossing  at  right  angles. 
Above  this  cross,  and  with  its  refracting  edge 
parallel  to  one  of  the  lines,  is  cemented  a 
small  glass  prism,  and  around  this  a  short 
section  of  a  glass  tube,  making,  thus,  a  shal- 
low vessel  with  a  prism  cemented  in  the 
bottom.  The  microscope  is  focussed  sharply 

upon  the  engraved  lines,  which  are  rotated  until  they  are  parallel  to  the 
cross-hairs.  The  fine  adjustment  screw  is  read,  after  which  the  ring  is 
filled  with  the  unknown  fluid.  It  will  be  found  that  the  engraved  lines 
are  now  displaced  a  certain  amount,  the  distance  depending  upon  the 
refractive  index  of  the  fluid,  and  in  order  to  bring  them  back  into  position, 
it  is  necessary  to  move  the  adjustment  screw.  By  making  determinations 
upon  a  series  of  fluids  of  known  indices,  a  curve  may  be  constructed  by  the 
aid  of  which  the  index  of  any  unknown  liquid  may  be  found,  using,  of  course, 
the  same  combination  of  ocular,  objective,  and  tube  length.  Clerici  claims 
the  method  to  be  accurate  to  the  third  decimal  place. 

1  Enrico  Clerici:  Sulla  determinazione  ddVindice  di  rifrazione  al  microscopic.  Rendi- 
conti  della  Reale  Accad.  dei  Lincei,  Roma,  XVI  (1907),  336-343. 


Fie.  363. — Clerici's  apparatus. 


CHAPTER  XV 
OBSERVATIONS  BY  ORDINARY  LIGHT  (Continued) 

DETERMINATION  OF  THE  REFRACTIVE  INDICES  OF  A  MINERAL 
BY  THE  BECKE  METHOD 

236.  Becke  (1893). — In  1893,  Becke1  called  attention  to  the  fact  that  at 
the  contact  between  two  transparent  minerals  of  different  refractive  indices 
in  thin  sections,  under  certain  conditions  of  illumination,  the  total  reflection 
of  some  of  the  rays  of  light  produces  a  characteristic  phenomenon. 

If  one  focusses  accurately,  with  a  medium-power  objective,  on  the  con- 
tact between  two  minerals  having  different  indices  of  refraction,  condenser 
and  analyzer  being  removed,  it  will  appear  as  a  sharp  line  when  it  lies  at 
right  angles  to  the  section.  If,  now,  some  of  the  light  entering  from  below 
be  cut  off  by  means  of  a  diaphragm,  and  the  tube  of  the  microscope  be  very 
slightly  raised  so  as  to  throw  the  image  somewhat  out  of  focus,  there  will 
appear  along  the  contact,  but  within  the  mineral  having  the  higher  index,  a 
bright  line  which  broadens  upon  raising  the  tube  still  farther  and  then  dis- 
appears. On  depressing  the  tube,  the  bright  line  appears  at  the  edge  of  the 
mineral  having  the  lower  index. 

The  phenomenon  observed  depends  upon  the  total  reflection  of  the  rays 
incident  at  more  than  the  critical  angle  when  passing  from  the  denser  to  the 
rarer  medium,  and  is  explained,  by  Becke,  by  means  of  the  illustration 
reproduced  as  Fig.  364. 

Let  AB  and  BC  be  two  minerals  in  contact  at  B,  and  let  the  refractive 
index  of  A  be  less  than  that  of  C.  Let  o  to  12  be  convergent  rays  of  light 
entering  from  below,  and  let  the  refraction  of  the  rays  above  and  below  the 
mineral  be  disregarded  since  this  is  of  no  importance  in  the  explanation. 

The  ray  o,  entering  the  minerals  perpendicular  to  their  surfaces,  suffers 
no  refraction  but  passes  straight  through.  The  rays  i,  3,  5,  7,  9,  and  n, 
travelling  from  the  rarer  medium  A  at  the  left,  to  the  denser  medium  C, 
are  bent  toward  the  normal,  and  pass  out  to  the  right.  The  rays  entering 
from  the  right,  however,  pass  from  a  denser  to  a  rarer  medium.  In  such 
cases  all  rays  impinging  on  the  second  at  more  than  the  critical  angle,  such 

1  F.  Becke:  Ueber  die  Bestimmbarkeit  der  Gesieinsgemengtheile,  besonders  der  Plagioklase, 
auj  Grund  Hires  Lichtbrechungsiermogcns.  Sitzb.  Akad.  Wiss.,  Wien,  CII  (1893),  Abth.  I, 
358-378. 

Idem:  Petrographische  Studien  am  Tonalit  der  Rieserferner.  Untersuchungsmethoden. 
T.  M.  P.  M.,  XIII  (1892-3),  385-389- 

271 


272 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  237 


as  2,  4,  and  6,  are  totally  reflected,1  and  emerge  upon  the  same  side,  while 
those  reaching  the  second  surface  at  a  smaller  angle  of  incidence,  such  as  the 
rays  8,  10,  and  12,  are  refracted.  In  consequence,  therefore,  of  the  total 
reflection  of  certain  rays,  the  light  is  unevenly  disturbed,  and  the  elevation 
of  the  objective  shows  the  concentration  of  the  light  on  the  side  of  the  mineral 
having  the  higher  refractive  index.  The  smaller  the  cone  of  entering  light, 
down  to  the  limit  of  the  critical  angle,  the  clearer  will  be  the  phenomenon 
observed,  for  if  only  the  rays  from  i  to  6,  in  the  figure,  enter  from  below, 
there  will  be  no  ray  passing  to  the  left  as  against  six  to  the  right.  The  less 
the  difference  between  the  indices  of  the  two  minerals,  the  greater  will  be 
the  critical  angle,  consequently  the  smaller  must  be  the  size  of  the  diaphragm 


024 


i3 15 


7       '          '          I  \          *          Q     ] 

53i0^468i 

FIG.  364-  FIG.  365. 

FIG.  364. — Becke's  explanation  of  the  bright  line  effect. 

FIG.  365. — Becke  line.  The  contact  between  the  two  minerals  is  inclined,  the  mineral  with 
lower  index  lying  above  the  other. 

FIG.  366. — The  Becke  line.  The  contact  between  the  two  minerals  is  inclined,  the  mineral  with 
higher  index  lying  above  the  other. 

used.     Based  upon  this,  there  was  proposed  by  Viola2  a  quantitative  measure 
of  the  refractive  indices. 

Becke  further  calls  attention  to  the  fact  that  there  is  no  difference  in  the 
phenomenon  observed  even  if  the  contact  between  the  two  minerals  is  not 
quite  vertical,  provided  the  medium  having  the  lower  index  lies  above  (Fig. 
365).  If,  however,  it  lies  below  (Fig.  366),  and  the  inclination  is  great 
enough,  the  bright  line  may  appear  to  move  the  wrong  way.  In  practice 
this  is  of  no  consequence  since  such  contacts  may  be  clearly  recognized, 
under  the  microscope,  by  the  shifting  of  the  line  when  the  focus  is  changed 
from  the  bottom  to  the  top. 

237.  Hotchkiss'  Explanation. — Hotchkiss3  gives  a  somewhat  different 

1  Art.  41  supra. 

2  Art.  241  infra. 

3  W.  O.  Hotchkiss:  An  explanation  of  the  phenomena  seen  in  the  Becke  method  of  deter- 
mining index  of  refraction.     Amer.  Geol.,  XXXVI  (1905),  305-308. 


ART.  237J 


OBSERVATIONS  BY  ORDINARY  LIGHT 


273 


explanation  as  follows:  Let  AB,  Fig.  367,  be  a  cross-section  of  two  minerals 
with  indices  of  1.50  and  1.70  respectively,  and  let  the  plane  of  contact  be 
perpendicular  to  the  page  and  represented  by  the  line  CD.  Let  the  conver- 
gent light  come  from  below  and  pass  through  the  section.  Ray  i  is  refracted 
so  as  to  meet  the  plane  between  the  two  media  at  the  point  n,  at  a  distance 
above  the  point  x  of  1.87  times  the  length  x-y.  Ray  2  meets  it  at  m,  a  dis- 
tance equal  to  1.56  times  x-y.  Ray  3  and  4,  since  B  has  the  higher  index, 
are  refracted  to  meet  the  plane  at  points  m'  and  n',  higher  than  the  similar 
rays  in  A,  or  at  distances  above  y  of  1.76  and  2.18  times  x-y,  respectively. 

At  the  surface  of  contact  between  A  and  B  the  critical  angle  is  62°  10', 
whereby  all  rays  incident  on  y-z  from  B  at  an  angle  greater  than  62°  10', 
are  totally  reflected  back  into  B.  On  the  other  hand,  a  portion  of  the  light 
from  A,  incident  upon  y-z,  is  refracted  into 
B.  The  ratio  between  the  amount  reflected 
and  the  amount  refracted  depends  upon 
several  factors.  In  proportion  as  the  con- 
tact surfaces  of  A  and  B  are  highly 
polished,  more  light  is  reflected  and  less 
refracted;  as  the  angle  of  incidence  in- 
creases, more  light  is  reflected  and  less 
refracted;  and  as  the  difference  in  the 
indices  increases  the  amount  of  light  re- 
flected* becomes  greater.  Since  the  con- 
tact surface  of  minerals  in  rocks  is  seldom 
smooth,  the  tendency  is  for  a  large  part  of 
the  light  from  A  to  be  refracted  into  B, 
and  the  condition  obtains  as  shown  in 

the  figure — that  for  a  certain  vertical  distance  along  the  contact,  approxi- 
mately equal  to  mn',  nearly  all  the  light  will  be  on  the  side  of  the  mineral 
having  the  higher  index.  If  the  microscope  is  focussed  within  this  vertical 
distance  a  band  of  light  will  be  seen.  If  the  tube  is  now  raised,  the  band 
will  be  seen  to  broaden,  as  is  evident  from  the  directions  of  the  refracted 
and  totally  reflected  rays.  If,  on  the  other  hand,  the  objective  is  lowered, 
the  band  becomes  narrower  and,  finally,  is  brighter  on  the  side  of  the  mineral 
having  the  lower  index,  which  is  explained  by  the  fact  that  the  light  in  A, 
which  is  approximately  the  same  in  amount  as  that  in  B  at  this  distance 
above  the  base  of  the  section,  is  concentrated  in  a  band  of  width  mr,  which 
is  shorter  than  ms,  and  will,  therefore,  show  greater  intensity. 

If  rays  from  B  are  incident  upon  y-z  at  an  angle  less  than  the  criitcal 
angle  (62°  10'  in  the  case  illustrated),  they  will  not  be  totally  reflected,  but 
will  partly  pass  through  into  A.  If  there  is  sufficient  light  thus  refracted, 
a  bright  band  will  be  seen  in  A  as  well  as  in  B  when  the  objective  is  raised. 
It  is  important,  therefore,  to  diaphragm  the  light  entering  the  condensing 

18 


FIG.   367. — Hotchkiss'  explanation  of  the 
Becke  line. 


274 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  238 


system  to  such  an  extent  that  all  the  light  from  B  is  totally  reflected  at  the 
contact  surface.  This  increases  the  relative  brightness  of  the  band  seen  in  B. 
Hotchkiss,  further,  computed  the  different  values  of  the  distances  from 
y  to  m  and  n  for  other  indices,  and  from  these  showed  how,  theoretically,  the 
indices  of  minerals  might  be  determined  from  the  differences  in  value.1 
Practically  the  magnitude  of  the  elevation  of  the  tube,  perhaps  0.0005  mm. 
for  the  change  from  1.54  to  1.56,  is  too  small  to  be  measured  accurately  with 
the  microscope. 

238.  Grabham's  Explanation. — In  the  previous  explanations  of  the 
Becke  line,  convergent  light  and  more  or  less  vertical  contact  was  necessary. 
An  explanation  based  on  parallel  rays,  such  as  are  ordinarily  used,  and 
inclined  junction  planes,  was  first  suggested  by  Anderson  to  Grabham.2 


11- 

— 

•E 

v 

1 

'1 

L- 

4- 

— 

12345878     9    10 
FlG.  371. 


FIG.  368.  FIG.  369.  FIG.  370. 

FIG.  368  TO  370. — Grabham's  explanation  of  the  Becke  line. 

When  the  plane  of  contact  between  the  two  minerals  is  inclined  to  that 
of  the  section,  two  cases  may  occur,  depending  upon  whether  the  mineral 
of  greater  or  less  refractive  index  overlaps  the  other.  In  the  former  case 
^  (Fig-  368)  the  rays,  coming  from  below,  pass  from 

a  rarer  to  a  denser  medium  and  are,  therefore,  bent 

toward  the  normal  at  the  point  of  contact.  Under 
the  second  condition  there  are  two  cases.  The 
rays,  passing  from  a  denser  to  a  rarer  medium, 
may  fall  upon  the  contact  at  more  than  the  critical 
angle  (Fig.  369)  and  be  totally  reflected,  or  they 
may  reach  it  at  a  less  angle  and  pass  through  but  will  be  bent  away  from 
the  normal.  In  any  case  the  light  is  increased  on  the  side  of  the  mineral 
having  the  higher  index  of  refraction. 

If,  now,  the  objective  is  focussed  upon  the  point  where  the  light  meets 
the  contact  (F,  Fig.  371),  the  section  will  appear  in  focus  and  no  bright  line 
will  be  seen.  If  the  tube  is  raised  so  that  the  focal  plane  is  at  H,  the  combi- 
nation of  ray  7  and  the  refracted  ray  6  produces  an  increase  of  light  at  that 
point.  If  the  tube  is  raised  still  farther,  the  point  of  light  moves,  progres- 
sively, farther  and  farther  toward  the  mineral  having  the  higher  index, 
according  as  each  vertical  ray  of  light  is  crossed  by  the  refracted  ray.  If, 

1  Cf.  C.  Viola:  Ueber  eine  neue  Methode  zur  Bestimmung  des  Brechungsiermogens  der 
Minerale  in  den  Dunnschlifien.    T.  M.  P.  M.,  XIV  (1894-5),  554-562.     See  Art.  240  infra. 

2  G.  W.  Grabham:  An  improved  form  of  petrological  microscope  with  some  general  notes 
on  the  illumination  of  microscopic  objects.     Mineralog.  Mag.,  XV  (1910),  341-347. 


ART.  239] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


275 


on  the  other  hand,  the  tube  of  the  microscope  is  lowered  so  that  the  focal 
plane  lies  at  L,  the  bright  line  will  appear  at  the  junction  of  ray  5  and  the 
backward  projection  of  the  refracted  ray  6.  Since,  under  the  microscope, 
ray  6  will  appear  to  come  from  a  point  on  its  dotted  backward  extension,  and 
not  from  the  point  6,  as  the  tube  is  lowered  more  and  more,  the  bright  line 
will  pass  progressively  through  its  intersections  with  rays  5,  4,  3,  etc. 

239.  Inclined  Illumination. — Becke1  called  attention  to  the  fact  that  the 
differences  between  the  indices  of  refraction  of  two  minerals  could  be 
brought  out  by  the  use  of  inclined  illumination.  For  this  purpose  he  dis- 
placed the  lower  diaphragm  laterally  and  found  that  the  edge  of  the  image, 
opposite  to  the  direction  in  which  the  diaphragm  was  displaced,  became  dark 
when  the  refractive  index  of  the  mineral  was  greater  than  that  of  the  adjacent 
one,  a  law  which  was  later  similarly  stated  by  Schroeder  van  der  Kolk.2 
Becke  produced  inclined  illumination,  likewise,  by  an  adaptation  of  theExner3 


FIG.  372. — The    Becke-Exner    mikrorefrac- 
tometer.     3/4    natural    size.     (Fuess.) 


FIG.  373. — The  Becke-Exner  mikrorefractom- 
eter.      ( Reich  ert.) 


microrefractometer.  This  instrument  is  shown,  in  the  simplified  form  sug- 
gested by  Becke,4  in  Fig.  372.  It  is  placed  on  the  end  of  the  tube  (T)  of 
the  microscope,  and  its  upper  part  is  extended  until  the  opening  of  the  dia- 
phragm lies  in  the  Ramsden  disk;  a  position  which  may  be  recognized  by 
the  disappearance  of  the  blue  halo  around  the  field.  The  disk  5,  which  cuts 
off  the  rays  from  one  side,  is  now  moved  laterally  across  the  opening,  by 
means  of  the  screw  K,  until  the  wished-for  effect  appears.  By  the  use  of  this 
instrument  a  difference  of  o.ooi  in  the  refractive  indices  of  two  adjacent 
minerals  may  be  recognized. 

Another  form  of  the  Becke-Exner  microrefractometer  is  shown  in  Fig. 
370.  This  instrument  is  attached  to  a  pivot  d,  so  that  it  may  be  swung  in  or 
out  of  the  field,  the  clip  b  holding  it  in  position  on  the  axis  of  the  microscope. 

1  F.  Becke:  Op.  ci!.,  Sitzb.  Akad.  Wiss.  Wien,  CII  (1893),  and  T.  M  P.  M.,  XIII  (1892-3), 

387. 

2  Art.  228. 

3  Sigm.   Exner:  Bin   Mikro-Refractomctcr.     Arch.   Mikrosk.   Anatomic,  XXV   (1885), 
97-112. 

4  C.  Leiss:  Die  optischcn  Instrnmcntc  dcr  Firm  a  R.  Fuess.     Leipzig,  1899,  246. 


276 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  240 


240.  Viola-de  Chaulnes-Becke  Method. — Viola,1  in  1895,  worked  out 
a  combination  of  the  de  Chaulnes  and  Becke  methods  by  means  of  which, 
theoretically,  the  refractive  indices  of  a  mineral  can  be  determined.2 

Let  M2  and  MI  (Fig.  374)  be  two  minerals  in  contact  whose  indices  of 
refraction  are  n%  and  HI,  respectively,  and  let  w2be  less  than  n\.  A  high- 
power  objective  is  focussed  on  the  upper 
surface  of  one  mineral,  and  the  microm- 
eter adjustment  of  the  microscope  is 
read.  The  objective  is  then  lowered, 
and  the  Becke  line,  on  the  side  of  the 
mineral  with  the  higher  refractive  index 
-M\  (Afi),  will  be  seen  to  become  gradually 
narrower  until  it  crosses  the  line  of  con- 
tact, which  will  be  in  sharp  focus  at  a\. 
The  micrometer  is  again  read,  the  differ- 
ence between  the  two  readings  giving  the 
value  of  e\.  The  distance  e  is  now  meas- 
ured, and  the  refractive  index  of  M\  de- 

FIG.  374. — Viola-de  Chaulnes-Becke 

method-  termined  by  the  equation  Wl  =  ?-.     The  in- 

dex of  the  mineral  (Mz)  having  the  lower  index  is  determined  by  measur- 
ing the  distance  e2  from  the  surface  to  the  point  where  the  dark  shadow  dis- 
appears and  the  bright  line  begins  to  show  on  the  side  of  M2.  Its  value  is 

found  from  the  equation  n2  =  --      The  writer  has  found  it  impossible,  in 

practice,  to  locate  the  positions  of  a\  and  #2  closely  enough  for  accurate 
determinations. 

241.  Viola-Becke  Method. — The  difference  between  the  refractive  indices 
of  two  abutting  minerals  in  a  thin  section,  according  to  Viola,3  are  pro- 
portional to  the  square  of  the  opening  of  the  iris  diaphragm  necessary  to 
see  the  Becke  line,  and  may  be  determined  by  the  formula 


where  n2  and  n\  are  the  indices  of  refraction  of  the  two  minerals,  D  the  di- 
ameter of  the  diaphragm  opening,  and  k  a  constant. 

If,  then,  the  refractive  index  of  one  mineral  is  known,  that  of  any  mineral 
in  contact  with  it  may  be  determined  by  measuring  the  greatest  diameter 
of  the  iris  diaphragm  at  which  the  Becke  line  is  visible.  The  constant  k 
may  be  determined  by,  means  of  two  known  substances.  For  example,  it 

1  C.  Viola:    Ueber  eine  neue  Methods  zur  Bestimmung  des  Brechungsvermogens  der  Min- 
erale  in  den  Dunnschlijfen.     T.  M.  P.  M.,  XIV  (1894-5),  554-562. 

2  Cf.  Hotchkiss,  Art.  237  supra. 

3  C.  Viola:    Methode  zur  Bestimmung  des  Lichtbrechungsvermogens  eines  Miner  ales  in 
den  Dunnschliffen.     T.  M.  P.  M.,  XVI  (1896-7),  150-154. 


ART.  242]  OBSERVATIONS  BY  ORDINARY  LIGHT  277 

may  be  measured  in  a  section  of  apatite  cut  at  right  angles  to  the  c  axis 
(o>=  1.638)  and  embedded  in  a  fluid  having  an  index  of  1.549.  The  objective 
is  first  sharply  focussed,  then  very  slightly  raised,  and  the  diaphragm  slowly 
closed  until  a  bright  line  appears  in  the  apatite  at  the  contact  with  the  fluid. 
D,  in  this  case,  Jet  us  assume,  was  15  mm.;  the  formula  becomes 

1.638—1.549 

2  =0.00039. 


As  an  example  of  measurement,  a  crystal  of  pyroxene  in  contact  with 
Canada  balsam  was  taken.  The  crystal  was  turned  until  its  7  direction  was 
parallel  with  the  vibration  direction  of  the  lower  nicol.  The  iris  diaphragm 
was  now  closed  until  the  Becke  line  appeared,  when  D  was  found  to  be  20.5 
mm.,  from  which  the  value  of  7  was  found  to  be  1.713.  The  section  was  now 
turned  90°  so  that  a  was  parallel  with  the  nicol.  Here  D=ig  mm.,  whereby 
a  =  1.690. 

Viola  claims  that  as  soon  as  the  eye  is  sufficiently  trained,  this  method 
is  accurate  to  the  third  decimal  place,  and  that  it  is  preferable  to  any  pre- 
viously discovered  method  on  account  of  its  simplicity,  and  because  it  is  not 
necessary  to  determine  the  thickness  of  the  section.  By  this  method  he 
determined  that  the  small  inclusions  in  certain  leucites  —  crystals  so  small  that 
no  interference  colors  were  shown  and  no  other  property  than  their  refractive 
indices  were  deterrninable  —  were  pyroxene  and  apatite. 

242.  Practical  Applications  of  the  Becke  Method.  —  The  Becke  method 
is  capable  of  wider  application  than  simply  to  determine  which  of  two  adja- 
cent minerals  has  the  higher  refractive  index.  If  that  of  one  mineral  is  known  , 
the  relation  of  the  other  to  it  is  known.  If  two  adjacent  minerals  are  known 
and  one  has  a  refractive  index  higher,  and  one  a  lower  than  the  unknown, 
there  are  established  definite  limits  for  the  unknown.  The  Becke  line1  may 
be  used  also  with  the  immersion  method.  It  is  more  sensitive  and  easier  to 
see  than  the  light  and  dark  borders  produced  by  inclined  illumination,  and, 
at  the  same  time,  it  may  be  seen  over  the  whole  field  of  the  microscope  at 
once.  It  makes,  for  example,  the  process  of  determining  the  relative  amounts 
of  orthoclase  and  quartz  in  a  fine  granular  groundmass  very  simple,  for  if 
the  objective  is  very  slightly  thrown  out  of  focus,  either  up  or  down,  the  two 
minerals  stand  out  clearly,  one  from  the  other.  No  special  preparation  of 
material  is  necessary,  no  attachments  not  ordinarily  provided  with  a  micro- 
scope are  needed,  and  no  elaborate  computations  are  required  to  obtain 
accurate  results.  A  series  of  tests,  made  by  de  Lorenzo  and  Riva,2  show  it 
to  be  accurate  to  ±0.001. 

1  The  term  Becke  line  was  proposed  by  W.  Salomon:  Ueber  die  Berechnung  des  vari- 
ablen  Werthes  der  Lichtbrechung  in  beliebig  orientirten  Schnitten  optisch  einaxiger  Mineralien 
von  bekannter  Licht-  und  Doppelbrechung.     Zeitschr.  f.  Kryst.,  XXVI  (1896).  182. 

2  G.  de  Lorenzo  and  C.  Riva:    Mem.  Ace.  Sci.  Napoli,  X  (1901),  1-60.*     Review  in 
Zeitschr.  f  Kryst.,  XXXV  (1902),  501-2. 


278 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  242 


The  measurement  of  the  refractive  indices  in  two  directions  in  ani so- 
tropic  crystals,  and  their  comparison  with  the  two  refractive  indices  of  some 
known  substance,  was  applied  by  Becke1  to  the  determination  of  the  feld- 
spars, and  was  found  to  give  good  results.  If  adjacent  sections  of  quartz  and 
feldspar,  with  extinction  directions  parallel,  are  chosen,  the  vibration  direc- 


AU   100  90 

An.    0  10 


0  j 

100  # 


1.590 

Albite 

Oligoclase 

A  ink-sine 

Labradorite 

Bytownite 

Anorthlte 

1.5'JO 

4. 

^ 

1^80 
1^70 

^ 

^ 

>^ 

^ 

^ 

^ 

^ 

^ 

^^• 

^ 

^ 

1.560 
1.550 
1.540 
1.530 

/ 

^- 

•^ 

0^, 

^ 

1.560 
e  of  Quartz 

1^50 

w  of  Qnartz 

1J40 

Canada  Bal 

1.530 

/^ 

K 

^ 

1 

^ 

^^ 

/ 

/ 

^ 

i. 

r- 

^ 

^ 

~S 

^ 

^ 

^ 

^ 

>' 

// 

/•" 

^ 

=0 

X 

/ 

_. 

^^ 

^ 

/ 

^ 

^y 

FIG.  375. — Curve  showing  the  refractive  indices  of  the  lime-soda  feldspars.     (Modified  from  Rosen- 

busch-Wulfing.) 

tions,  necessarily,  must  be  parallel  also,  consequently  if  the  sections  happen 
to  be  cut  along  the  maximum  and  minimum  directions  (known  by  their 
maximum  interference  colors),  co  and  e  will  be  parallel  or  at  right  angles  to 
a  and  7.  Taking  the  values  of  a,  /?,  and  7  of  the  plagioclases  and  o>  and  e  of 
quartz  as  given  by  Rosenbusch,  the  relationships  are  shown  by  Fig.  375, 
from  which  the  following  grouping  is  derived: 


Group 

Parallel  position       At  right  angles 

Feldspars 

I.                    co>«       6>T              o;>T       e>a 

Albite, 

Ab  —  AngAni 

II.                    to>a       e>7              <*>5*7       e>« 
III.                    co^a       e>7              co<7       c>« 

>  Oligoclase, 

lAbsAmiAb'An; 

IV.                    co<tt       e  ^7              co<7       €>a 
V.                    a><oj       c<7              co<7       e  ^  a 

|  Andesine, 

/  Ab2Ani  —  AbsAn2 
1  AbsAn2  —  AbiAni 

1  f  Labradorite,    1 

VI. 

co<a    •  c<7             u><7       6<a         j  Bytownite, 
(  Anorthite. 

>  AbiAni  —  An 

J 

From  this  it  appears  that  the  calcic  plagioclases  always  have  higher  refract- 


1  F.  Becke:    Op.  cit.,  Akad.  Wiss.  Wien,  CII  (1893),  and  T.  M.  P.  M.,  XIII  (1892-3), 
387-396. 


ART.  242] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


279 


ive  indices  than  quartz,  and  albite  and  oligoclase-albite  always  lower.     The 
other  plagioclases  may  be  separated  from  each  other  as  shown. 

But  not  only  may  quartz  be  used,  but  any  other  known  mineral  as  well, 
for  example,  nephelite.  Potassium  feldspars  and  anorthoclase  have  all  their 
refractive  indices  lower  than  e  of  nephelite;  the  plagioclases  bear  to  it  the 
following  relations.1 


sTephelite:  co  =  1.542,  6  =  i  .  537.) 

In 
parallel 
position 

At 

right 
angles 

Albite, 
Oligoclase-albite, 

co  >7     6  >  a. 

| 
/  co>a     e  ^7 

Oligoclase, 

co  ^7 

co  ^  a 

Andesine, 

1 

] 

Labradorite, 

Ico 

( 

•  t  ~>y 

Bytownite, 

CO 

€<a 

>  co  <a 

Anorthite, 

J 

\ 

0    €=  1.5533 


<•=  1.5522 


A  drawback  to  this  method  of  Becke  is  that  the  sections  of  quartz  used 
for  comparison  must  be  cut  approximately  parallel  to  cry  stall  ographic  c,  or 
the  values  of  o>  and  c  will  not  be  at 
their  maxima.  If  those  sections  are 
chosen  which  give  high  interference 
colors,  as  was  suggested  by  Becke, 
they  may  still  not  be  parallel  to  c  but 
may  show  that  the  section  is  thicker 
than  normal.  Salomon2  overcame  this 
difficulty  by  calculating  the  maximum 
value  of  the  extraordinary  ray  for  any 
section  of  quartz  (Fig.  376)  by  means 
of  the  equation 


eco 


1.5495 


r  1.5465 


(Eq.  ga,Art  53), 

where  v  is  the  angle  between  the  sec-  FlG  3?6 

tion  and  the  optic  axis.     To  determine 

the  angle  c/>,  Salomon  made  use  of  interference  figures  and  measured  the 

amount  of  inclination  of  the  optic  axis  from  the  axis  of  the  microscope. 

While  the  description  of  the  method  is  in  advance  of  the  description  of  inter- 

1  Albert  Johannsen :  Determination  of  rock-forming  minerals.     New  York,  1908,  76. 

2  W.  Salomon:    Ueber  die  Berechnung  des  lariablen  Wertes  der  Lichtbrechung  in  beliebig 
orientirten  Schnitten  optisch  einaxiger  mineralien  von  bekannter  Licht-  und  Doppelbrechung. 
Zeitschr.  f.  Kryst.,  XXVI  (1896),  178-187. 


280 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  242 


ference  figures,  it  is,  nevertheless,  inserted  here  as  the  most  convenient  place 
for  reference. 

Four  cases  occur. 

1.  The  center  of  the  interference  figure  lies  within  the  field  of  view. 

2.  The  center  of  the  interference  figure  lies  beyond  the  field  of  view  but 
the  bars  are  sharp  enough  to  permit,  with  sufficient  accuracy,  the  measure- 
ment of  their  distances  from  the  center  of  the  field. 

3.  Similar  to  case  2,  but  the  bars  are  too  indistinct  to  permit  of  accurate 
measurements. 

4.  The  section  shows  the  characteristic  figure  of  a  section  cut  approxi- 
mately parallel  to  crystallographic  c. 

In  the  first  case,  with  the  aid  of  a  Bertrand  lens  and  a  micrometer  ocular, 
the  distance  between  the  emergence  of  the  optic  axis  and  the  center  of  the 
field  is  measured.  The  displacement  is  reduced  to  the  true  value  of  <p  by 
means  of  Mallard's  formula.1 

kd 

sin  <p  =  — ,  (i) 

CO 

in  which  k  is  a  constant  for  the  particular  combination  of  lenses,  tube  length, 
etc.,  used,  and  d  the  measured  distance.  The  constant  should  be  determined 
previously  by  making  a  number  of  measurements  on  substances  whose 
values  of  co  and  <p  are  known. 

In  the  second  case,  the  distance  between 
centers  may  be  determined  in  the  following 
manner.  The  stage  is  first  carefully  centered 
and  is  then  rotated  until  one  of  the  black  bars 
(OC,  Fig.  377)  passes  exactly  through  the  cen- 
ter O.  The  vernier  is  read  at  this  position, 
after  which  the  stage  is  rotated  until  the  same 
bar  passes  through  D,  the  limit  of  the  field  of 
view.  Obviously  the  point  of  emergence  of  the 
FlG.  377-  optic  axis  has  moved  from  A  to  B,  about  O  as  a 

center  and  with  a  radius  of  OA  =  OB  =  d,  and 
through  an  angle  0.     If  BC  =  OD  =  r,  the  radius  of  the  field  of  view,  then 

Substituting  in  equation  (i),  we  have 


r  17  Y 

sin  p  —  -;.  and  d=  . — _• 
a  sin  a 


sin  <p  =  — —0. 
co  sin  p 


(2) 


The  accuracy  of  the  measurements,  in  this  case,  depends  upon  the  sharp- 
ness of  the  axial  bars,  which  themselves  increase  in  distinctness  with  increas- 
ing strength  of  double  refraction,  thickness  of  section,  and  convergence  of 
rays  from  the  condenser. 


1  Art.  411  infra. 


ART.  242] 


OBSERVATIONS  BY  ORDINARY  LIGHT 


281 


In  the  third  case,  it  is  not  possible  to  determine  the  value  of  <p  accurately. 

In  the  fourth  case,  the  dark  bars  are  generally  too  indefinite  for  the 
determination  of  how  nearly  the  section  is  parallel  to  the  optic  axis.  If 
exactly  parallel,  the  value  of  <p,  of  course,  is  o°. 

All  measurements,  in  the  above  determinations,  should  be  repeated  a 
number  of  times  to  insure  accuracy. 

In  Becke's  method  for  the  determination  of  the  plagioclase  feldspars  they 
are  separated  into  six  groups;  by  Salomon's  method  they  may  easily  be  sepa- 
rated by  means  of  their  birefringence,  into  nine  groups,  each  differing  from 
the  next  by  o.ooi.  This  would  correspond  to  sections  of  quartz  differing 
by  10°  in  their  inclination  to  crystallographic  c,  whereby  errors  of  less  than 
5°  would  make  but  slight  differences  in  the  results  obtained. 

The  computed  values  of  €1  for  various  angles  of  inclination  are  given  in 
the  table  below  and,  graphically,  in  Fig.  378. 


<f> 

«i 

<p 

*1 

Q0° 

•5442 

40° 

-5495 

80° 

•5445 

30° 

•5510 

70° 

•5453 

20° 

•5522 

60° 

•5465 

10° 

•5530 

SO" 

.5480 

0° 

•5533 

1.553 
1.552 
1.501 
1.550 
1.549 
1.548 
1.547 
1.546 
1.545 
1.544 

"^ 

\ 

[Maximum    1 

\ 

V^ljies  of  c{,   1 

\ 

in 

9 

ct^ois  inc 

linec 

^ 

It 

any  angle 

(4 

l) 

\ 

w 

th 

trie 

)pl 

ic 

\ 

ixis 

i 

\! 

\ 

' 

1 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

"^ 

FIG.  378. —  Maximum  values  of  «i  in  sections   in- 
clined at  any  angle  (<p)  with  the  optic  axis. 


Fr.om  these  determinations  the 
value  of  the  birefringence  (c-co)  for 
any  particular  section  may  be  ob- 
tained, and,  from  that,  the  thickness 
of  the  section  may  be  accurately  de- 
termined. 

A  second  drawback  to  the  method 
of  plagioclase  determination,  as  pro- 
posed by  Becke,  is  that  in  a  rock  section  in  which  there  is  but  little  quartz  and 
feldspar,  few  or  no  cases  may  be  found  in  which  adjacent  grains  of  the  two 
minerals  extinguish  simultaneously.  The  value  for  the  refractive  index  of 
quartz  in  a  direction  parallel  to  the  vibration  directions  of  the  unknown 
mineral  may,  however,  be  computed,  and  thus  permit  the  use  of  any  section. 

When  a  plane  polarized  ray  of  light  reaches  a  section  of  quartz  cut  in  any 
direction  except  at  right  angles  to  the  optic  axis,  it  is  broken  up  into  two  rays. 
These  two  rays,  the  extraordinary  and  the  ordinary,  on  leaving  the  quartz, 
emerge  at  different  points,  it  is  true,  but,  from  some  other  rays  of  light 
impinging  at  a  slightly  different  point,  there  will  likewise  arise  ordinary  and 
extraordinary  rays,  so  that,  with  a  beam  of  light  entering  from  below,  from 
every  point  of  the  upper  surface  of  the  quartz,  two  rays  having  different 
indices  will  emerge.  If,  then,  the  vibration  direction  of  the  polarizer  is  not 


282 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  242 


parallel  to  one  of  the  principal  vibration  directions  of  the  quartz,  the  refract- 
ive index  seen  by  the  eye  is  not  the 
true  but  only  an  apparent  index,  and 
consists  of  a  combination  of  the  two. 

c<H^—  — ^P  c^Jn  —±P  For  example,  let  CP,  Fig.  379,  rep- 
resent the  amount  and  direction  of  the 
vibration  of  the  light  entering  the 
quartz  from  the  polarizer,  and  let  the 
vibration  directions  of  the  latter  lie 


FIG.  380. 


exactly  at  45°  to  the  former. 
Then,  by  the  resolution  of  forces, 
CE  and  CO  will  represent  the  ex- 
traordinary and  ordinary  com- 
ponents into  which  this  ray  is 
broken  by  the  quartz.  Since 
the  amplitudes  and  intensities  of 
the  two  rays  are  here  equal,  the 
resulting  apparent  refractive  in- 
dex will  be  just  half  that  of  the 
ordinary  ray  alone  plus  half  that 
of  the  extraordinary  alone,  or 

— .     If,    however,    the  angle 


=1.554 


1.553 


0°      10°      20°      30°      40°      50°      60°     70°      80°     90°= 


1.544, 


Uo 


ECP  has  a  different  Value,  repre-    FIG.  381.— Diagram  giving   values   of   nq   for  all  values 

sentedbyp,  we  have  (Fig.  380), 

CE 


That  is,  the  cotangent  of  p  shows 
the  relation  between  the  extra- 
ordinary and  the  ordinary  ray, 
so  that  if  there  were  one  part  or- 
dinary ray  entering  into  the  re-, 
suit,  there  would  be  cotan  p 
parts  of  extraordinary  rays. 
The  apparent  refractive  index 
(nq)  will  no  longer  be  the  arith- 
metical mean  between  co  and  ci, 
but  will  be  the  mean  of  one 
ordinary  ray  with  an  index  co, 

FK,  ^.^^.p^^a^owta.  the  same   ^       CQtan        p       extraordinary 

rays  with   index  of  ei,  whereby 
the  mean  index  would  be  represented  by  the  equation : 


ART.  243] 


OBSERVATIONS  BY  ORDINARY  LIGHT 

oj+€i  co tan  p 
Hq~   i+cotan  p 


283 


From  this  equation  the  apparent  index  of  refraction  in  any  section  of 
quartz  may  be  computed,  consequently  it  is  not  necessary,  in  order  to  deter- 
mine the  refractive  index  of  an  unknown  mineral,  that  the  sections  of  quartz 
should  have  extinctions  parallel  with  it.  The  proportion  of  usable  sections  in 
a  rock -slice  is  thus  greatly  increased. 

Computed  by  this  formula,  the  following  values  of  nq  were  obtained. 

Values  for  nq  for  various  values  of  p. 


P=o°  
p—  10° 

•  5442 

C442 

i  -  5445 

1-5453 

1.5465 

1.5480 

r-5495 

I-55IO 

I-5522 

1.5530 

1-5533 

I     CtJIQ 

0  —  20° 

ZA.A.2 

cco8 

p  =  22  1/2°  

n  —  2O° 

•5442 
1:4.4.2 

1.5458 

i  .  5469 

i.548o 

i  -  5490 

i  -  5499 

•5506 
5^00 

P       6    o  

—  4O 

CAA2 

•  ^492 

n  =  4  r° 

$44.2 

i    <?444 

i   5448 

I    C4C4 

i  .  5461 

I  .  N460 

I   .   ^476 

i  .  5482 

I  .  5486 

.5488 

P  =  10    .  . 

C442 

•  5484 

0  —  6O 

C442 

^47^ 

—  6?   1/2° 

C442 

I    ^4^3 

I    <?4s8 

I       ^462 

I    e>46? 

^460 

—  7O° 

C442 

^467 

0  =  80° 

^442 



^4^6 

0=00°. 

C442 

I    ^442 

I     ^442 

I     ^442 

I    ^442 

I    ^442 

I    =J442 

I  .  5442 

I  .  5442 

.  ^442 

<t>  = 

90° 

80° 

70° 

60° 

50° 

40° 

30° 

20° 

10° 

0° 

They  are  shown  graphically  in  Fig.  381,  and  stereographically  in  Fig.  382. 
The  former  figure  clearly  brings  out  the  fact  that  at  45°  the  value  for  ng  is  the 
mean  between  p  =  go°  and  p  =  o°. 

The  above  method  may  be  used,  not  only  with  quartz,  but  with  any  uni- 
axial  mineral  if  the  proper  values  are  computed.  It  makes  possible  the 
determination  of  the  refractive  index  at  every  contact,  no  matter  how  the 
crystal  may  be  oriented,  between  an  unknown  mineral  and  a  uniaxial  crystal 
of  known  refractive  index;  the  directions  chosen  for  determination  in  the 
unknown  being  naturally  along  the  principal  vibration  directions. 

243.  Refractive  Index  of  Canada  Balsam. — One  of  the  most  convenient 
standards  with  which  to  compare  the  refractive  index  of  an  unknown  mineral 
in  a  thin  section  is  the  medium  in  which  it  is  mounted,  usually  Canada 
balsam.  While  this  substance  is  amorphous,  unfortunately  its  index  is  not 
absolutely  constant  but  varies  with  the  method  of  preparation,  the  amount  of 
heating  during  the  process  of  mounting  the  rock-slice,  and  the  age  of  the 
preparation,  especially  if  air  has  had  access  to  it.  The  variations,  however, 
are  too  small  to  be  taken  into  consideration  for  most  minerals,  and  it  is  only 
for  minerals  whose  indices  of  refraction  fall  within  the  limits  of  variation  of  the 
balsam,  that  accurate  determinations  are  necessary. 

Canada  balsam  has  been  used  as  a  means  of  comparison  ever  since  the 


284  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  243 

Becke  method  came  into  use  in  1893,  but  it  was  only  recently  that  extensive 
determinations  of  its  refractive  index  and  its  variation  have  been  made.  The 
values  given  in  the  older  works  differ  decidedly,  in  most  cases  being  given 
much  too  high.  Brewster1  gave  1.549,  Behrens2  1.528-1.540,  Klein3  1.536, 
Zirkel4  1.549,  Becker5  1.5393,  Rosenbusch-Wiimng  ±i-546  and  1.542  to 
i.55o.7  In  1909  Calkins8  compared  the  index  of  Canada  balsam  with  vari- 
ous minerals  in  300  thin  sections  from  one  to  eight  years  old,  and  found  that 
in  only  one  case  out  of  a  hundred  did  the  index  of  balsam  exceed  1.544.  The 
lowest  value  obtained  was  between  1.535  ±0.002.  He  gives  1.54  as  a  fair 
mean  value,  and  says  the  refractive  index  is  rarely  less  than  1.535  nor  more 
than  1.545.  Schaller9  made  a  number  of  determinations  with  a  refrac- 
tometer,  on  blank  slides  prepared  for  the  purpose,  and  found  that  uncooked 
balsam  in  sodium  light  has  an  index  of  1.524,  soft-cooked  an  average  of 
1.5387,  as  usually  cooked  in  the  mounting  of  thin  sections  1.5377,  and  over- 
cooked an  average  of  1.5412  with  a  maximum  value  of  1.543.  Wiilnng10 
made  determinations  by  comparison  with  minerals  in  thin  sections  prepared 
thirty  to  forty  years  previously  and  also  with  an  Abbe-Pulfrich  total  refrac- 
tometer  on  fresh  balsam  obtained  from  six  different  firms.  He  found,  in  a 
collection  thirty  years  old,  that  the  central  portions  had  a  value  of  1.538  + 
0.002,  while  the  borders,  which  had  become  yellow  with  age,  averaged  1.5416. 
Other  sections  gave  values  between  1.5330  and  1.5382,  the  mean  being 
1.537  ±0.004.  He  concluded  that  the  index  in  the  majority  of  the  slides  of 
the  Heidelberg  collection  lies  between  1.533  and  1.541,  and  only  in  rare  cases 
does  it  reach  1.544  or  fall  below  1.533,  both  cases  being  due  to  fault  of  manu- 
facture. Balsam  which  has  turned  yellow  does  not  always  have  a  high  index, 
but  all  balsam  when  exposed  to  air  discolors,  becomes  brittle,  and  increases 
in  index.  Balsam  protected  by  a  cover-glass  or  by  a  crust  of  balsam  may 
retain  its  sticky  consistency  and  low  index  even  for  forty  years;  it  therefore 

1  Sir  David  Brewster:    A  treatise  on  new  philosophical  instruments.     Edinburgh,  1813. 
Book  IV,  Chapter  II  tables. 

2  Wm.  Behrens:    Tabellen  zum  Gebrauch  bei  mikroskopischen  Arbeiten.     Braunschweig, 
i  Aufl.,  1887,  Tabelle  XXVII. 

3  Carl  Klein:    Ueber  die  Methode  der  Einhiillung  der  Krystalle  zum  Zweck  ihrer  optischen 
Erforschung  in  Medien  gleicher  Brechbarkeit.     Neues  Jahrb.,  1891  (I),  70-76. 

4  F.  Zirkel:    Lehrbuch  der  Petrographie.    I,  2te  Aufl.,  Leipzig,  1893,  40. 

6  G.  F.  Becker:  Reconnaissance  of  the  gold  fields  of  southern  Alaska.  18  Ann.  Rep. 
U.  S.  Geol.  Survey,  pt.  Ill,  Washington,  1898,  30.  Determination  of  the  refractive  index 
of  balsam  by  Prof.  J.  E.  Wolff. 

6  Rosenbusch-Wiilfing:    Mikroskopische  Physiographic,  I\,  4te  Aufl.  1904,  150. 

7  Idem:  Ibidem,  1 2,  .345. 

8  F.  C.  Calkins:    Refractive  Index  of  Canada  balsam.     Science,  N.  S.  XXX  (1909),  973. 

9  Waldemar  T.  Schaller:    The  refractive  index  of  Canada  balsam.     Amer.  Jour.  Sci., 
XXIX  (1910),  324. 

10  E.  A.  Wiilfing:  Ueber  die  Lichtbrechung  des  Kanadabalsams.     Sitzb.  Akad.   Wiss. 
Heidelberg,  Math.-naturw.  Kl.,  1911,  20  Abhandl.,  1-26. 


ART.  245]  OBSERVATIONS  BY  ORDINARY  LIGHT  285 

is  altered  only  on  the  surface  or  at  the  borders.  Commercial  balsams  are  so 
uniform  that  in  the  preparation  of  thin  sections  the  limiting  values  of  the 
index  need  not  fall  outside  the  limits  1.533  and  1.541,  and,  with  practice, 
should  be  between  1.534  and  1.540. 

244.  Relation  between  Refractive  Index  and  Density. — Various  formulae 
have  been  empirically  determined  to  express  the  relation  between  the  re- 
fractive indices  of  substances  and  their  densities,  the  simplest  one  being  that 
of  Gladstone  and  Dale1  which  is, 


where  K  is  a  constant. 

A  formula,  determined  independently  by  Lorentz2  and  Lorenz,3 

«2—  i     i 


is  more  complex  than  that  of  Gladstone  and  Dale  and,  according  to  Larsen,4 
no  more  accurate,  one  formula  holding  as  well  as  the  other. 

245.  The  Examination  of  Opaque  Minerals. — While  opaque  minerals  are 
of  comparatively  slight  importance  in  ordinary  petrographic  work,  they 
are  of  great  importance  in  the  study  of  ore  deposits.  Until  within  a  com- 
paratively recent  period,  no  serious  attempts  were  made  to  study  them 
microscopically.  With  the  development  of  microscopical  metallographic 
methods,  however,  the  possibility  of  studying  opaque  minerals  by  the  same 
means  was  opened  up,  and  while  the  methods  are  even  now  not  fully  de- 
veloped, what  has  been  done  is  sufficient  to  indicate  the  possibilities.  The 

1  J.   H.   Gladstone  and  T.  P.  Dale:  Researches  on  the  refraction,  dispersion,  and  sensi- 
tiveness of  liquids.     Phil.  Trans.  Roy.  Soc.,  London,  CLIII  (1863),  317-343,  especially  320. 

2  H.   A.  Lorentz:  Ueber  die  Beziehung  zu'ischen  der  Fortpflanzungsgeschwindigkeit  des 
Lichles  und  der  Korperdichte.     Wiedem.  Ann.,  IX  (1880),  641-665. 

3L.  Lorenz:    Ueber  die  Refractionsconstante.     Ibidem,  XI  (1880),  70-103. 

4  Esper  S.  Larsen:  The  relation  between  the  refractive  index  and  the  density  of  some  crys- 
tallized silicates  and  their  glasses.  Amer.  Jour.  Sci.,  XXVIII  (1909),  263-274. 

See  also 

H.  L.  Barvlr:  Ueber  die  Verhdltnisse  zunschen  dem  Lichtbrechungsexponenten  und  der 
Dichte  bei  einigen  Miner  alien.  Sitzb.  Gesell.  Wiss.  Prag,  1904,  No.  3.* 

F.  Slavik:  Review  of  above  in  Zeitschr.  f.  Kryst.,  XLII  (1905-6),  410-411. 

M.  Sprockhoff:  Beitrdge  zu  den  Beziehungen  zu'ischen  dem  Kry stall  und  seinem  chem- 
ischen  Bestand.  Neues  Jahrb.,  B.  B.,  XVIII  (1903-4),  117-154. 

Michael  Stark:  Zusammenhang  des  Brechungsexponenten  natiirlicher  Gldser  mit  ihrem 
Chemismus.  T.  M.  P.  M.,  XXIII  (1904),  536-550. 


286  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  245 

scope  of  this  work  is  too  limited  to  insert  these  methods  here,  and  the  student 
is  referred  to  the  papers  mentioned  below.1 

Another  method,  which  promises  to  be  of  value  for  the  study  of  opaque 
minerals,  is  that  of  staining  employed  by  Leo,2  who  gives  the  characteristic 
colors  produced  on  a  limited  number  of  minerals. 

1  Joh.    Koenigsberger:   Zur  optischen  Bestimmung  der  Erze.     Centralbl.  f.  Min.,  etc., 
1901,  195-197. 

William  Campbell:  The  microscopic  examination  of  opaque  minerals.  Econ.  Geol.,  I 
(1905-6),  751-766. 

W.  Campbell  and  C.  W.  Knight:  A  microscopic  examination  of  the  Cobalt  nickel  arsen- 
ides and  silver  deposits  of  Temiskaming.  Ibidem,  767-779. 

Wm.  Campbell  and  C.  W.  Knight:  On  the  microstructure  of  nickeliferous  pyrrhotites. 
Ibidem,  II  (1907),  350-366. 

Joh.  Konigsberger:  Ueber  einen  Apparat  zur  Erkennung  und  Messung  optischer 
Anisotropie  undurchsichtiger  Substanzen  und  dessen  Verwendung.  Centralbl.  f.  Min.,  etc., 
1908,  565-573,  597-605.  Translation  in  Winchells'  Elements  of  optical  mineralogy.  New 
York,  1909,  465-475. 

Francis  Church  Lincoln:  Certain  natural  associations  of  gold.  Econ.  Geol.,  VI  (1911), 
247-302. 

L.  C.  Graton  and  Joseph  Murdoch:  The  sulphide  ores  of  copper.  Trans.  Amer.  Inst. 
Mining  Eng.,  New  York  meeting,  Feb.,  1913,  741-809. 

2  Max  Leo:    Die  Anlaujfarben.  Eine  neue  Methode  zur  Untersuchung  opaker  Erze  und 
Erzgemenge.     Dresden,  1911,  68  pp. 


CHAPTER  XVI 
MEASUREMENTS  UNDER  THE  MICROSCOPE 

246.  Measurement  of  Enlargement.  —  It  has  already  been  shown1  that 
the  magnifying  power  of  a  microscope  is  represented  by  the  equation 

*-•-¥• 

consequently  it  may  be  computed  from  this  equation  or  from  the  known 
magnifying  powers  of  ocular  and  objective.2  If  these  are  unknown,  it  may 
be  determined  by  direct  comparison,  with  a  scale,  of  the  magnified  image  of 
an  object  of  known  size. 

Upon  the  stage  of  the  microscope  is  placed  a  so-called  object-micrometer, 
which  consists  of  a  thin  glass  slide  upon  which  there  has  been  engraved  or 
photographed  a  millimeter  divided  into  ten  or  a  hundred  parts.  After  care- 
ful focussing,  the  image  seen  through  the  microscope  is  compared  with  an 
ordinary  millimeter  scale  which  is  placed  alongside  the  microscope  at  right 
angles  to  the  axis  of  the  instrument  and  at  a  distance  of  250  mm.  (the  distance 
of  distinct  vision)  from  the  exit  pupil.  If  one  now  observes  the  microscopic 
image  with  one  eye  and  the  scale  with  the  other,  by  shifting  the  scale,  the 
two  may  be  made  to  appear  to  lie  together,  and  the  enlargement  determined. 
For  example,  if  25  divisions  (0.25  mm.)  of  the  object  micrometer  correspond 
to  70  divisions  (70  mm.)  of  the  scale,  the  enlargement,  in  diameters,  for  that 
particular  combination  of  ocular,  objective,  and  tube  length,  will  be 


Instead  of  using  both  eyes,  a  camera  lucida  may  be  employed,  and  the 
length  of  a  certain  part  of  the  object-micrometer  may  be  drawn  on  a  sheet 
of  paper  placed  at  a  distance  of  250  mm.  from  the  eye.  Care  must  be  taken 
to  tilt  the  microscope  to  the  proper  angle  to  give  an  undistorted  image,  the 
amount  depending  upon  the  kind  of  camera  lucida  used.  The  distance, 
250  mm.,  must  be  measured  from  the  eye  point,  consequently  it  must  be  the 
actual  length  of  the  path  of  the  rays,  traced  through  all  the  angles  of  its 
reflection  in  the  camera  lucida.  The  line  traced  upon  the  paper  is  measured, 
finally,  and  the  computation  made  as  before. 

247.  Measurement  of  the  Field  of  View.  —  By  an  application  of  the  measure 
of  enlargement,  the  comparative  values  of  the  fields  of  view  of  different 
oculars  may  be  made.  The  apparent  diameter  of  the  field  at  250  mm.  is 

1  Arts.  98  and  165. 

2  \V.  Le  Conte  Stevens:  Microscope  magnification.     Amer.  Jour.  Sci.,  XT.  (1890),  50-62. 

287 


288  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  248 

determined  by  comparison  with  a  scale,  or  a  circle  is  drawn,  with  the  aid  of  a 
camera  lucida,  around  the  periphery. 

248.  Measurement  of  Lengths. — The  actual  size  of  a  microscopical  object 
may  be  determined  in  several  ways.  With  a  microscope  fitted  with  a  micro- 
meter stage,  one  may  determine  dimensions  directly  by  making  first  one  side 
'of  an  object  coincide  with  the  cross-hairs,  and  then  the  other.  The  difference 
between  the  two  readings  of  the  vernier  is  the  required  length.  This  is  the 
quickest  method  of  measurement  and,  with  some  stages,  readings  to  0.0005 
mm.  are  possible. 

More  accurate  measurements  may  be  made  by  means  of  micrometer 
oculars.  They  are  of  two  kinds,  scale-micrometer  oculars  and  screw-micro- 
meter oculars.  Scale-micrometers  may  be  attached  to  Huygens,  Ramsden, 
or  compensating  oculars.  In  the  Seibert  Huygens  scale-micrometer  ocular, 

shown  in  Fig.  383,  the  casing  is  made 
to  unscrew  in  the  middle,  so  that  a 
micrometer  scale  M ,  "shown  alone  at  the 
right,  may  be  placed  within  it.  The 
position  of  the  rabbet  is  such  that  when 
the  scale  is  inserted  with  the  engraved 
side  uppermost,  the  latter  coincides  with 
the  image  formed  by  the  microscope, 
and  is  magnified  with  it  by  the  eye-lens. 
By  drawing  out  the  eye-lens,  more  or 

FIG.  383. — Scale  micrometer  ocular.     (Seibert.)         J 

less,  the  micrometer  divisions  may  be 
brought  sharply  into  focus. 

Since  the  scale,  so  inserted,  is  magnified  by  the  ocular,  it  must  be  cali- 
brated for  each  different  combination  of  ocular,  objective,  and  tube  length. 
This  may  be  done  by  placing  an  object-micrometer  upon  the  stage  and 
noting  the  number  of  divisions  of  the  former  corresponding  to  a  certain 
number  of  the  latter.  For  example,  if  five  divisions  of  the  ocular-scale 
correspond  with  one  of  an  object-scale  which  is  divided  into  o.oi  mm.,  then 
one  division  of  the  former  corresponds  to  0.002  mm.  (2.0/z).1  Instead  of 
determining  the  number  of  divisions  corresponding  with  a  single  division  of 
the  object- micrometer,  it  is  better  to  choose  a  larger  number,  since  it  reduces 
the  error  of  the  determination. 

In  the  Zeiss  compensating  oculars,  which  are  used  with  apochromatic 
objectives,  the  divisions  of  the  scale  are  so  calculated  that  with  a  tube  length 
of  1 60  mm.  each  division  is  almost  exactly  equal  to  as  many  microns  as  there 
are  millimeters  in  the  focal  length  of  the  objective.2 

1  A  micron,  represented  by  /*,  is  a  thousandth  of  a  millimeter.     fi/J.  is  a  millionth  of  a 
millimeter. 

2  S.  Czapski:  Compensationsocular  6  mil   i/i   Mikron-Theilung  zum  Gebrauch  mil  den 
apochromatischen  Objectiven  von  Carl  Zeiss  in  Jena.     Zeitschr.  f.  wiss.  Mikrosk.,  V 
I50-I5S. 


ART.  248] 


MEASUREMENTS  UNDER  THE  MICROSCOPE 


289 


A  variety  of  the  scale-micrometer  ocular  is  the  net-,  coordinate-,  or  cross- 
grating-micrometer  ocular.  This  differs  only  from  the  preceding  in  having 
the  glass  plate  engraved  with  cross-section  lines  (Fig.  384)  instead  of  with 
a  simple  scale.1  In  the  ocular  shown  in  Fig.  383  the  glass  scales  are 
interchangeable. 

For  still  more  accurate  results,  a  screw-micrometer  ocular2  may  be 
used  (Fig.  385).  Between,  or  beneath,  the  lens-combination,  as  the  case 
may  be,  depending  upon  the  type  of  eyepiece  used,  is  fitted  a  scale,  usually 
marked  with  0.5  mm.  divisions.  Immediately  above  or  below  this  plate 
is  another,  marked  with  a  single 
line,  and  capable  of  being  moved 
along  the  former  by  means  of  a 
micrometer  screw.  A  complete  rev- 
olution of  the  drum  in  most  screw- 
micrometer  oculars  moves  the  scale 
0.5  mm.;  with  100  divisions  upon  it, 
each  one  indicates  a  movement  of 


0.005    mm- 


Like   the  scale- 


FIG.  384. — Net  grating  for  micrometer 
ocular.      (Fuess.) 


FIG.  385. — Screw-micrometer  ocular.     (Zeiss.) 


micrometer  ocular,  this  also  must  be  calibrated  by  means  of  a  stage-mi- 
crometer, and  the  number  of  divisions  of  the  drum  corresponding  to  one 
division  of  the  scale  determined. 

1  C.  Leiss:  Mittheilungen  aus  der  R.  Fuess' schen  Werkstatte.     Ocular  zur  Mes sung  der 
Mengenverhdltnisse  verschiedener  Miner  ale  in  einem  Dunnschliff.     Neues  Jahrb.,  1898  (II), 
70. 

2  Hugo  von  Mohl:  Ueber  eine  neue  Einrichtung  des  Schraubenmikrometers.     Arch.   f. 
mikrosk.  Anat.,  I  (1865),  79-100. 

Alfred  Koch:  Eine  Combination  von  Schraubenmikrometer  und  Glasmikro  meter  ocular. 
Zeitschr.  f.  wiss.  Mikrosk.,  VI  (1889),  33-35. 

W.  A.  E.  Drescher:  New  accessories  of  the  Bausch  &  Lomb  Optical  Company.  Filar 
micrometer.  Proc.  Amer.  Microsc.  Soc.,  i2th  Ann.  meeting,  Buffalo,  XI  (1889),  132-133. 
(Describes  a  screw  micrometer  ocular  after  the  designs  of  M.  D.  Ewell.) 

Anon:  Bulloch's  improved  filar  micrometer.    Jour.  Roy.  Microsc.  Soc.,  1891,  106-107. 

For  lost  motion  see  V.  Knorre:  Untersuchungen  uber  Schraubenmikrometer.     Zeitschr. 
f.  Instrum.,  XI  (1891),  41-5°,  83~93- 
19 


290  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  249 

To  determine  the  size  of  an  object,  one  side  is  made  to  coincide  with 
some  mark  on  the  scale,  the  full  divisions  between  this  and  the  other  side  are 
counted,  the  fractional  part  remaining  is  measured  by  the  micrometer 
screw,  and  the  true  value  computed.  If  the  object  is  small,  the  movable 
index  mark  may  be  made  to  coincide,  successively,  with  its  two  sides,  and 
the  number  of  revolutions  of  the  micrometer  screw  noted.  Like  in  an  ordi- 
nary eyepiece,  the  eye-lens  is  movable  to  permit  of  accurate  focussing  upon 
the  scale.  The  instrument  is  inserted  in  the  draw  tube  of  the  microscope 
and  is  rigidly  clamped  by  means  of  the  screw  at  the  side. 

The  ocular  shown  in  Fig.  386 
was  designed  by  Wright1  for  the 
special  purpose  of  measuring  axial 
angles,  though  it  may  be  used  for 
all  the  purposes  to  which  the  pre- 
ceding can  be  put.  In  the  place  of 
a  movement  in  one  direction  only, 
this  ocular  has  two  movements  at 
right  angles  to  each  other.  Its  use 
will  be  discussed  more  fully  below. 

FIG.  386.- Wright's  double  screw-micrometer  Measurement    Of     AieaS.— 

Ramsden  ocular.      1/2  natural  size.     (Fuess.) 

The  principal  purpose  for  which  areas 

are  measured  in  petrographic  work  is  the  determination  of  the  volume  per- 
centage of  the  constituents  of  rocks.  One  of  the  earliest  methods  proposed 
was  that  of  Delesse,2  which  is  based  on  the  assumption  that  the  sum  of  the 
areas  of  each  of  the  constituents  in  a  section  of  a  uniformly  homogeneous 
rock  is  proportional  to  the  actual  volume  of  that  constituent.  His  method 
was  to  make,  first,  a  drawing  of  each  constituent  in  the  rock  by  tracing 
carefully,  on  thin  oiled  paper,  the  outlines  of  each  mineral  as  shown  in  a  pol- 
ished slab.  Each  kind  of  mineral  was  then  differently  colored  in  the  draw- 
ing, and  the  whole  was  pasted  on  a  piece  of  tin  foil,  after  which  it  was  care- 
fully cut  apart  on  the  lines.  The  different  colors  were  now  carefully  sorted, 
the  tissue  paper  and  gum  were  soaked  off,  and  the  tin  foil  was  weighed  for 
each  constituent. 

1  Fred.  Eugene  Wright:  The  measurement  of  the  optic  axial  angle  of  minerals  in  the  thin 
section.     Amer.  Jour.  Sci.,  XXIV  (1907),  336. 

Idem:  Das  Doppel-Schrauben-Mikrometer-Okular  und  seine  Anwendung  zur  Messung 
des  Winkels  der  optischen  Achsen  ion  Krystalldurchschnitten  unter  dem  Mikroskop.  T.  M. 
P.  M.,  XXVII  (1908),  299. 

Idem:  The  methods  of  petrographic-microscopic  research.  Carnegie  Publication  No. 
158,  Washington,  1911,  155. 

2  A.   Delesse:  Procede  mechanique  pour  determiner  la  composition  des  rochcs.     Comp- 
tes  Rendus,  XXV  (1847),  544-545.     Brief  of  following. 

Idem:  Same  title.     Ann.  d.  Mines,  XIII  (1848),  379-388. 


ART.  249] 


MEASUREMENTS  UNDER  THE  MICROSCOPE 


291 


1     2 


78  9  10  11  12  13  14  15  16  17  18  19  29  21 


Sollas1  improved  this  method  by  making  his  drawing  by  means  of  a 
camera  lucida. 

A  somewhat  similar  method  was  used  by  Joly2  for  determining  the  pro- 
portions of  hard  and  soft  constituents  in  paving  material.  Instead  of  using 
a  camera-lucida,  he  made  use  of  a 
photographic  apparatus,  and 
traced  with  ink  the  outlines  of  any 
particular  constituent  upon  the 
back  of  a  photographic  plate  upon 
whose  front  side  a  positive  of  co- 
ordinate paper  was  printed.  Upon 
holding  the  transparent  plate  to 
the  light,  the  number  of  square  mil- 

riG.  387. — Comparison  of  linear  measurements  with 

limeters  or  square  centimeters  con-  total  areas. 

tained  within  the  ink  outlines  could  ™ 

be  estimated.     The  whole  circular  area  being  equal  to    : — ,  the  area  occupied 

by  the  mineral  could  be  estimated  as  a  percentage  of  that  of  the  field,  several 

drawings  being  made  and  an  average  taken  of  all. 

Rosiwal3  still  further  improved  the 
method  by  reducing  his  measurements  to 
linear  series  in  two  directions.  His  method 
is  based  on  the  principle  that  the  total  length 
of  all  measured  lines,  as  a  to  k  and  i  to  21 
(Fig.  387),  bears  the  same  relation  to  the 
portions  intercepted  on  these  lines  by  each 
constituent,  as  the  volume  of  the  whole  rock 
does  to  that  of  each  constituent.  His  actual 
method  of  procedure  was  to  draw  rectangular 
coordinates  upon  the  cover-glass,  to  add  to- 
gether all  the  intercepts,  and  finally  to  com- 
pare them  with  the  total  length  of  line?, 
measured. 

Hirschwald4  simplified  the  measurement 

1  W.  J.  Sollas:  Contributions  to  a  knowledge  of  the 
granites  of  Leinster.     Read  Nov.  30,    1889.     Trans. 
Roy.  Irish  Acad.,  Dublin,  XXIX  (1887-1892),  427-512,  in  particular  471-473. 

2J.  Joly:    The  petrological  examination  of  paiing-sets.     Proc.   Roy.   Dublin  Soc.,  X 
(1903-5),  62-92. 

3  August  Rosiwal:     Ueber  geometrische  Gesteinsanalysen.     Ein  einf acker  Weg  zur  zif- 
fernmassigen   Feststellung   des   Quantitatsverhaltnisscs   der   Miner albestandtlieile   gemengter 
Gesteine.     Verh.  d.  k.  k.  geol.  Reichsanst.,  Wien,  1898,  143.* 

4  J.  Hirschwald:    Ueber  ein  neuer  Mikroskopmodell  und  ein  "Planitneter-Ocular"  zur 
geometrischen  Gesteinsanalyse.     Centralbl.  f.  Min.,  etc.,  1904,  626-633. 

Idem:  Handbuch  der  bautechnischen  Gestehisprilfung.     I,  Berlin,  1911,  146-147,  163-172. 


FIG. 


388. — Hirschwald's      planiraeter 
ocular.     (Puess.) 


292 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  249 


by  the  use  of  his  planimeter  ocular  (Fig.  388).  This  consists  of  a  Huygent, 
ocular,  in  the  focal  plane  of  which  there  are  two  glass  micrometer  scales, 
10  mm.  long,  and  perpendicular  to  each  other.  Of  the  two  scales,  one  is 
movable,  in  a  direction  at  right  angles  to  its  length,  by  means  of  a  milled 
head,  the  other  is  stationary.  This  ocular  is  much  more  convenient  to  use 
than  a  net  micrometer  ocular  (Fig.  384)  since  one  is  much  less  likely  to  lose 
sight  of  the  place  under  count.  It  is,  however,  less  convenient,  though 
more  rapid,  than  an  ordinary  screw-micrometer  ocular. 

To  test  the  accuracy  of  this  method  of  measuring  the  area  of  any  con- 
stituent in  a  rock,  Hirschwald,  in  a  manner  similar  to  that  already  used 
by  Rosiwal,  cut  a  piece  of  paper,  10  sq.  cm.  in  area,  into  small  irregular  pieces, 
and  pasted  them,  haphazard,  upon  a  quadratic  ruled  sheet  of  50  sq.  cm.  area 
(Fig.  387).  The  exact  proportion  of  the  irregular  portion,  as  compared  with 
tlie  larger  sheet,  was  20  per  cent. ;  as  determined  by  the  linear  measurements, 
it  was  411  to  2000,  or  20.6  per  cent. 

This  method  of  determining  areas  by  linear  measurements,  generally 
spoken  of  as  the  "  Rosiwal  method,"  is  very  convenient  for  determining  the 
composition  of  a  granular  rock.  If  the  individual  components  are  of  known 
composition,  the  complete  analysis  of  the  rock  can  be  computed.  The  linear 
measurements  are  first  reduced  to  100,  the  values  thus  representing  the 
relative  volume  of  each  component.  The  volumes  are  then  multiplied  by 
the  specific  gravity  of  the  corresponding  mineral,  and  the  total  again  reduced 
to  100,  to  give  the  percentage  weights,  or  masses,  of  each. 

The  following  mechanical  analysis  of  the  "Butte  granite,"  given  by 
Cross,  et  al.,1  may  be  taken  as  an  example. 

MECHANICAL  ANALYSIS  OF  THE  "BUTTE  GRANITE" 


Total 
diameters 

Relative 
volumes 

Sp.  gr. 

Weights 

Quartz  
Orthoclase  .  .  . 

2,954 
I  373 

23.17 
18  62 

2.65 

2     ^7 

22.55 

2  1    O7 

Plagioclase  

t  4.O2 

A'Z      IO 

2  68 

4.2    4.7 

Biotite  

I   I3O 

8  87 

300 

977 

Hornblende 

482 

•2       7g 

Pyroxene  .  . 

2^2 

I    O7 

•2    -20 

2     37 

Magnetite  .    . 

e  i 

c  40 

51  7 

o  76 

Pyrite  

6 

O    O4. 

SOO 

o  07 

12,740 

99-95 

99.98 

In  this  analysis  the  12,740  units  of  the  micrometer  scale  represent  the  total 
distance  measured  and  are  the  sum  of  604  grains  traversed.  An  idea  may 
hereby  be  obtained  of  the  number  of  readings  necessary. 

1  Cross,  Iddings,  Pirsson,  Washington:  Quantitative  classification  of  igneous  rocks. 
Chicago,  1903,  226. 


ART.  251] 


MEASUREMENTS  UNDER  THE  MICROSCOPE 


293 


250.  Measurement    of    Thicknesses. — Thicknesses    may  be  measured 
by  means  of  the  fine  adjustment  screw  of  the  microscope.     The  instrument 
should  first  be  carefully  focussed  on  a  scratch  on  an  object  slip,  after  which 
the  mineral  to  be  measured  should  be  placed  above  the  mark,  by  sliding  it 
over  to  exclude  the  air,  and  its  upper  surface  brought  into  focus.     The  differ- 
ence between  the  micrometer  readings  gives  the  measure  of  the  thickness.     In 
order  to  correct  any  lost  motion  which  may  be  present  in  the  screw,  the 
microscope  should  be  brought  into  focus,  in  both  cases,  by  turning  the  screw 
in  one  direction  only. 

If  the  measurement  is  to  be  made  by  focussing  through  the  mineral,  the 
method  of  the  Due  de  Chaulnes1  may  be  used,  whereby  D  =  nM  (Eq.  2, 
Art.  208),  D  being  the  true  thickness,  M  the  measured  thickness,  and  n 
the  index  of  refraction  of  the  mineral.  In  using  this  method  the  meas- 
urements should  be  made  near  the  center  of  the  field,  otherwise  the  curva- 
ture of  the  image  may  produce  a  considerable  error. 

A  much  more  delicate  measure  of  thickness  is  by  means  of  the  birefrin- 
gence of  a  mineral.2 

251.  Measurement  of  Plane  Angles. — In  measuring  plane  angles  under 
the  microscope,  it  must  be  remembered  that  the  apparent  angle  of  cleavage3 
may  not  be  the  true  angle.     Wherever  possible, 

the  section  for  measurement  should  be  so  chosen 
that  the  planes,  whose  intersection  is  to  be  meas- 
ured, lie  parallel  to  the  axis  of  the  microscope. 
When  this  occurs,  the  junction  line  between  the 
two  will  not  be  displaced  upon  focussing  succes- 
sively upon  the  bottom  and  top  of  the  slide. 

In  making  a  measurement,  the  apex  of  the 
angle  is  set  on  the  intersection  of  the  cross-hairs,  and  one  leg  is  made  to 
coincide  with  one  of  them.  The  stage  vernier  is  now  read  and  the  stage 
rotated  until  the  other  leg  of  the  angle  coincides  with  the  same  cross-hair. 
The  difference  between  this  reading  and  the  former  gives  the  angle.  In- 
stead of  making  the  edges  of  the  mineral  and  the  apex  of  the  angle  coincide 
exactly  with  the  cross-hairs,  whereby  a  slight  angle  may  be  concealed  by  the 
thickness  of  the  hairs,  it  will  be  found  better  to  place  them  just  a  trifle  to 
one  side,  the  parallel  position  being  determined  by  the  uniform  width  of 
the  hair-line  of  light  between  the  two. 

A  method  which  is  especially  useful  for  small  crystals  is  measurement 
with  the  aid  of  a  Leeson  prism.4  This  instrument,  called  a  double  refracting 

1  See  also  Art.  208,  supra. 
-  Art.  301,  infra. 

3  Art.  206,  supra. 

4  H.  B.  Leeson:    On  crystallography,  with  a  description  of  a  new  goniometer  and  crystal- 
lonome..     Mem.  and  Proc.  Chem.  Soc..  London,  III  (1848),  486-560,  in  particular  550-552. 


FIG.  389. — Leeson  prism. 


294 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  251 


goniometer  by  the  inventor,  depends  on  the  action  of  a  doubly  refracting 
prism,  either  of  Iceland  spar  or  of  quartz,  of  such  thickness  that  it  will  only 
partially  separate  the  two  images  of  the  angle  which  is  to  be  measured.  It 
may  be  used  as  a  separate  instrument,  or  attached  to  a  microscope,  or  even 
attached  to  a  telescope  to  measure  the  dip  of  strata.  Used  on  a  microscope, 
it  is  slipped  over  the  ocular,  after  the  manner  of  a  cap  nicol,  and  the  amount  of 
rotation  is  read  from  a  vernier.  It  is  shown  in  section  in  Fig.  389  in  which  a 


FIG.  390. 


FIG.  391. 


FIG.  392. 


is  an  achromatic  prism  of  Iceland  spar  or  a  Rochon  quartz  prism,  b  a  sliding 
collar  for  adjustment,  v  the  vernier,  and  Oc  the  ocular  of  the  microscope. 
When  a  crystal  or  a  cleavage  angle  is  viewed  through  the  prism,  polarizer  and 
analyzer  of  the  microscope  being  removed,  two  images,  somewhat  separated, 
will  appear.  Upon  rotating  the  cap,  the  extraordinary  image  will  revolve 
about  the  ordinary,  and  will  occupy  various  positions  as  shown  in  Figs.  390 
to  392. 


FlG.  393. — Ocular  goniometer.     3/4 
natural  size.     (Fuess.) 


FIG.  394. — Ocular  goniometer.     (Reichert.) 


Let  abc,  Fig.  390,  be  the  angle  to  be  measured.  The  vernier  is  set  at  o° 
and  clamped,  and  the  tube  containing  the  prism  is  revolved  until  the  lines 
forming  one  side  of  the  angle  to  be  measured  coincide  in  both  images  (ab  and 
a'b',  Fig.  391).  The  vernier  is  now  released  and  the  whole  instrument  is 
revolved  on  the  graduated  scale  until  the  two  lines  forming  the  other  side  of 
the  angle  coincide  (be  and  b'c',  Fig.  392).  The  amount  of  rotation  is  the 
measure  of  the  angle  or  of  its  complement  according  to  the  direction  in  which 


ART.  252]  MEASUREMENTS  UNDER  THE  MICROSCOPE  295 

the  prism  was  revolved.  It  is,  of  course,  not  at  all  necessary  to  set  the 
vernier  at  o°  for  the  first  reading;  the  difference  between  the  two  is  sufficient. 

Another  method  of  measuring  plane  angles  is  by  means  of  an  ocular  go- 
niometer (Fig.  393).  A  diaphragm  e,  adjusted  by  means  of  four  centering 
screws  i,  carries  on  its  upper  edge  a  single  cross-hair  passing  exactly  through 
the  axis  of  the  microscope.  Above  this  the  diaphragm  /,  likewise  adjustable 
by  centering  screws,  carries,  on  its  lower  edge,  another  cross-hair,  also  pass- 
ing accurately  through  the  axis.  By  means  of  the  screw  c,  the  two  are 
brought  as  nearly  as  possible  together,  in  which  position  the  two  hairs  lie 
in  the  focal  plane  of  the  Ramsden  ocular  above.  To  read  an  angle,  one  leg 
is  first  placed  parallel  to  the  lower,  and  stationary,  cross-hair,  after  which  the 
upper  cross-hair  is  rotated  until  it  coincides  with  the  other  leg.  The  angle 
may  be  read  to  minutes  by  means  of  the  vernier  n.  Another  form  of  ocular 
goniometer  is  shown  in  Fig.  394. 

If  one  uses  a  microscope  with  a  cap  nicol  simultaneously  rotating  with 
the  polarizer,  consequently  with  an  ocular  likewise  rotating  at  the  same  time, 
one  may  read  the  angle  to  within  5  minutes  by  the  vernier  there  provided. 

252.  Measurement  of  Optic  Axial  Angles. — The  measurement  of  optic 
axial  angles  is  discussed  below.1 

1  Chapters  XXXIV-XXXV. 


CHAPTER  XVII 


DRAWING  APPARATUS 

253.  Drawing  Apparatus.  —  All  drawing  instruments  for  use  with  the  mi- 
croscope are  based  on  the  principle  of  the  camera  lucida.  The  drawing 
paper  and  the  image  seen  through  the  microscope  appear  to  lie  superimposed, 
in  the  same  plane,  due  to  reflection  through  a  prism  ;  consequently  it  is  an 
easy  matter  to  make  a  drawing  of  a  rock  section  by  tracing  the  outlines. 

The  simplest  kind  of  drawing  apparatus  is  in  the  form  of  a  single 
prism  which  may  be  permanently  attached  to  an  ocular  (Figs  395-396)  or 
removable  (Fig.  397).  Such  prisms  are  of  two  types.  In  one1  the  edge 
of  the  prism  just  reaches  the  axis  of  the  microscope 
(Fig.  395)  and  but  half  the  light  is  used.  The  rays 
of  light,  coming  from  the  drawing,  meet  the  lower 
surface  of  the  prism  at  right  angles  and,  after 
being  twice  reflected,  emerge  at  right  angles  to  the 
upper  surface.  With  an  instrument  as  shown  in  Fig. 
395  the  image  is  projected  close  to  the  stand  of 
the  microscope.  In  an  improved  form,  shown  in 
Fig.  396,  the  prism  is  so  modified  that  when  the 
microscope  is  inclined  45°,  the  image  appears,  with- 
out  distortion,  on  the  horizontal  surface  back  of  the 
stand.  The  light  from  the  image  and  from  the  paper 
mav  ke  equaiize(i  by  means  of  two  tinted  glasses 
which  swing  on  a  pivot  below  the  prism. 
The  Nachet2  camera  lucida  (Figs.  397-398)  is  simple  and  satisfactory.  It 
is  so  constructed  that  the  light  from  the  whole  field  passes  through  the 
prism.  This  is  accomplished  by  cementing  a  small  triangular  prism  to  one  of 
the  faces  of  a  rhombic  prism  so  that  the  light,  coming  from  the  image,  strikes 
the  lower  face  at  right  angles,  passes  through  without  refraction,  and  emerges 
at  right  angles  to  the  upper  surface.  The  rays  from  the  drawing  likewise 
enter  and  emerge  at  right  angles  to  the  faces,  but  suffer  two  reflections  in  their 
course.  The  image  seen  with  this  camera  lucida  is  of  the  entire  field  of  view 

1  P.  Schiemenz:  Die  neue  Zeichenoculare  von  Leitz.     Zeitsch.  f.  wiss.  Mikrosk.,  XII 
(1895),  289-292. 

2  Anon:  Nachet's  improved  camera  lucida.     Jour.  Roy.  Microsc.  Soc.,  II  (1882),  260- 
261.     The  same  instrument  is  described  as  Swift's  by  Frank  Crisp:  On  some  recent  j  or  ms 
of  camera  lucida.     Jour.  Roy.  Microsc.  Soc.,  II  (1879),  21-24. 

296 


FIG.     395. — Section 
drawing  apparatus. 


through 


ART.    2o:* 


DKAU'fXG  APPARATUS 


297 


and  it  is  undistorted  when  the  microscope  is  vertical  and  the  drawing  table 
at  right  angles  to  the  line  of  projection.     A  blue  glass  neutralizes  part  of  the 


Fie.  396. — Improved  drawing  apparatus.     (Leitz.) 

light  coming  from  the  drawing  and  permits  the  image  to  be  clearly  seen, 
another  form1  the  prism  is  so  cut  that  the  microscope  may  be  inclined. 


In 


L\ 


FIG.  397. — Camera  lucida.     (Xachet.) 


FIG.  398. — Passage  of  light  through 
the  Nachet  camera  lucida. 


The  Abbe  type  of  drawing  apparatus2  (Fig.  399)  consists,  of  two  triangular 
prisms,  silvered  on  the  plane  of  contact  between  them  with  the  exception 

1  Anon:  Nachefs  camera  lucida.     Jour.  Roy.  Microsc.  Soc.,  VI  (1886),  1057. 

2  S.  Czapski:  Ueber  einen  neuen  Zeichenapparat  und  die  Construction  ion  Zeichenappar- 
aten  im  attgemeinen.     Zeitschr.  f.  wiss.  Mikrosk.,  XI  (1894),  289-298. 

Anon:  Directions  for  using  the  Abbe  drawing  apparatus.     Zeiss'  circular  Mikro    118, 
pp.  8,  Jena,  1911. 


298 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  253 


of  a  small  opening  on  the  axis  of  the  microscope.  The  light  from  the  object 
passes  through  this  opening  while  the  images  of  the  drawing  paper  and  pencil 
are  reflected  by  a  mirror  and  by  the  silvered  surface  of  the  prism.  The  light 
from  the  paper  may  be  moderated  by  one  or  more  tinted  glasses,  and  the 


FIG.  399. — The  Abbe  drawing  apparatus.     2/3  natural  size.      (Zeiss.) 

whole  instrument  may  be  tilted  out  of  the  way  on  a  hinge.  When  the  mirror 
is  set  at  45°,  a  position  indicated  by  a  stop  in  the  instruments  of  some  makers, 
and  the  microscope  is  placed  in  an  upright  position,  the  image  appears  un- 
distorted  upon  the  paper.  The  part  of  the  drawing  nearest  the  microscope, 


FIG.  400. — Tilting  drawing-board.     (Bausch  and  Lomb.) 

however,  is  cut  off  by  the  foot  and,  if  the  entire  field  is  to  be  drawn,  it  is 
necessary  to  change  the  inclination  of  the  mirror.  This  introduces  an 
error,  however,  for  when  the  inclination  of  the  mirror  to  the  drawing-board 
is  not  exactly  45°,  a  distortion  is  produced  in  the  image,  and  it  is  necessary 
to  correct  this  by  means  of  a  tilting  drawing-board  such  as  is  shown  in  Fig. 


ART.  253]  DRAWING  APPARATUS  299 

400. l  This  may  be  inclined  in  various  directions  and  securely  clamped  in 
such  positions.  It  may  be  raised  or  lowered,  also,  in  order  to  modify  the 
size  of  the  image  produced.  When  the  drawing-board  lies  at  a  distance  of 
250  mm.  from  the  exit  pupil  of  the  microscope,  the  distance  being  measured 
through  all  the  changes  of  path  of  the  ray  in  passing  through  the  drawing 
apparatus,  the  image  will  be  drawn  with  the  so-called  magnification  of  the 
microscope. 

The  drawing-board  shown  in  the  figure  is  24  by  37  1/2  cm.,  and  has  an 
adjustable  arm  rest  on  the  front  edge.  The  base  is  28  by  51  cm. 

To  test  the  accuracy  of  the  setting  of  a  camera  lucida2  use  may  be  made 
of  test  objects,  such  as  circles,  squares,  or  parallel  lines  drawn  on  a  glass  slip. 
The  drawings  made  from  such  objects  may  be  measured:  a  compass  set  in 
the  center  of  the  circle  should  exactly  follow  the  lines  drawn,  the  square 
should  have  equal  sides,  and  the  parallel  lines  be  truly  parallel. 

1  For  other  forms  see  Wilhelm  Bernhard:  Ein  Zeichentisch  fur  mikroskopische  Zwecke. 
Zeitschr.  f.  wiss.  Mikrosk.,  IX  (1892),  439-445. 

Idem:  Zusatz  zu  meinem  Aiijsatz  "Ein  Zeichentisch,  etc."     Ibidem,  298-301. 
Dr.  Giesenhagen:  Ein  Zeichenpult  fur  den  Gebrauch  am  Mikroskop.     Zeitschr.  f.  wiss. 
Mikrosk.,  VII  (1890),  169-172. 

2  J.  Anthony:  On  drawing  prisms.     Jour.  Roy.  Microsc.  Soc.,  IV  (1884),  697-703. 


CHAPTER  XVIII 
ROTATION  APPARATUS 

254.  Rotation  Apparatus. — Under  the  heading  of  rotation  apparatus  are 
included  here  all  those  appliances,  accessory  to  a  microscope,  by  which 
crystals  or  thin  sections  may  be  tilted  from  the  horizontal  so  that  the  axis  of 
the  microscope  passes  through  them  at  different  angles  than  before.  While 
the  principal  use  of  rotation  apparatus  is  for  the  examination  of  minerals  by 
polarized  light,  yet  for  certain  purposes,  such  as  the  measurement  of  inter- 
facial  angles  of  small  crystals,  or  of  cleavage  angles,  they  are  used  in  ordinary 
Ugh'. 

Probably  the  first  rotation  apparatus  used  with  a  microscope  was  that 
invented  by  Leeson1  in  1848,  and  used  by  him  to  tilt  crystals  into  proper 
positions  for  measuring  angles.  It  had  three  movements,  two  of  them  hori- 
zontal and  one  vertical,  whereby  a  crystal  could  be  turned  to  any  position  in 
two  planes. 

The  next  instrument,  eight  years  later,  by  Highley,2  was  not  a  detachable 
stage,  but  a  built-in  part  of  an  inverted  chemical  microscope.  Instead  of  the 
ordinary  revolving  stage,  this  instrument  carried  two  concentric  graduated 
rings,  one  of  which  had  the  usual  movement  in  azimuth,  while  the  other  was 
pivoted  and  had  a  movement  in  altitude. 

Valentin,3  in  1861,  used  an  apparatus  which  could  be  clamped  to  the  stage 
of  the  microscope.  It  consisted  of  a  rotating  disk  attached  to  an  arm  by 
which  it  had  a  movement  in  altitude.  No  graduated  circle  was  provided, 
consequently  there  was  no  means  of  measuring  the  amount  of  rotation. 

The  rotating  stage  described  by  Nageli  and  Schwendener,4  while  an 
improvement  on  the  two  preceding,  was  not  as  complete  as  Leeson's.  It 
consisted  of  a  horizontal  plate  and  a  vertical  graduated  circle.  The  move- 
ment was  about  a  horizontal  axis,  the  amount  being  indicated  by  a  pointer 
without  a  vernier.  A  suggestion  was  made  that  for  certain  purposes  it  would 
be  advisable  so  to  arrange  the  apparatus  that  it  could  be  rotated  in  a  trough 
under  water  or  some  other  liquid. 

1  H.  B.  Leeson:  On  crystallography,  with  a  description  of  a  new  goniometer  and  crystal- 
lonome.     Mem.  and  Proc.  Chem.  Soc.  London,  III  (1848),  486-560.     In  particular  550- 

552- 

2  Samuel  Highley:  Contributions  to  micro-mineralogy.     Quart.  Jour.  Microsc.  Soc.,  IV 
(1856),  277-286. 

3  G.  G.  Valentin :  Die  Untersuchung  der  Pflanzen  und  Thiergewebe  in  polarisirtem  Lichte. 
Leipzig,  1 86 1,  166.* 

4  Carl  Nageli  und  S.  Schwendener:  Das  Mikroskop.    Leipzig,  i  Aufl.,  1867,  2.  Aufl., 
1877.     English  translation,  The  Microscope,  New  York,  2nd  ed.,  1892,  315-319- 

300 


ART.  254]  ROTATION  APPARATUS  301 

An  appliance  resembling  the  preceding  was  constructed  by  von  Ebner1 
in  1874.  The  whole  instrument  was  attached  to  the  glass  bottom  of  a  trough, 
75  mm.  by  35  mm.  and  18  mm.  deep,  which  could  be  filled  with  an  immersion 
fluid.  The  amount  of  rotation,  about  50°  each  way,  was  indicated  by  a 
pointer,  without  vernier,  and  was  necessarily  only  approximate. 

A  simple  rotating  stage,  without  graduated  scales,  was  made  by  West.2 
Bertrand,3  in  1880,  described  an  instrument  very  similar  to  that  of  Nageli  and 
Schwendener,  except  that  the  graduated  half  circle  was  outside  the  immersion 
trough  and  that  there  was  a  vernier  attached  to  the  end  of  the  pointer.  A 
modern  instrument,  constructed  by  Fuess4  on  this  principle,  is  shown  in  Fig. 
401.  The  specimen  is  held  in  the  pin- 
cette P  or  on  an  object  carrier  O,  the 
latter  consisting  of  a  glass  slip  and  a 
forfc-shaped  spring.  S  is  a  screw  and 
Sch  a  sliding  bar  by  means  of  which  the 
holder  P  is  brought  into  the  axis  of  ro- 
tation. To  Bertrand's  instrument 
there  was  the  objection  that  the  hori- 
zontal axis  penetrated  the  side  of  the 
vessel  containing  the  immersion  fluid, 
which  made  it  almost  impossible  to  pre- 
vent the  escape  of  some  of  the  latter 

.    ,  .  ._,,  .      FIG.  401. — Small  rotation  apparatus.     2/3  nat- 

upon  the  stage  of  the  microscope.     This  ura,  size.    (Fuess.) 

is  overcome,  in  the  Fuess  apparatus,  by 

the  horseshoe-shaped  piece  i    which,   however,   only  permits   a  rotation 

through  125°.     The  circle  is  graduated  to  degrees. 

In  1884  Brogger5  described  an  instrument  which  he  inserted  in  the  central 
opening  of  the  microscope  and  used  in  orienting  small  crystals  for  goniome- 
tric  measurements.  It  is  shown  in  section,  twice  the  size  of  the  original, 
in  Fig.  402.  A  plate  d,  which  rests  upon  the  stage  of  the  microscope,  carries  a 
disk  a  into  which  fits  the  hemisphere  c.  The  latter  is  drilled  out  in  the  center 
and  in  it  is  placed  the  table  b,  upon  which  the  crystal  to  be  examined  is  placed; 
and  which  may  be  raised  or  lowered.  A  modified  form  of  this  instrument, 
which  may  be  used  for  other  purposes  as  well,  is  constructed  by  Fuess,6  and 
is  shown  in  Fig.  403.  On  a  round  disk,  which  is  attached  to  the  stage  of  the 

1  V.    von  Ebner:    Untersuchungen   iiber  das  Verhalten  des  Knochengewebes  im  polar- 
isirten  Lichte.     Sitzb.   Akad.  Wiss.  Wien,  Math.-naturwiss.  Kl.,  LXX  (1874),  iii  Abth. 
111—115. 

2  Anon:  West's  universal-motion  stage  and  object  holder.     Jour.  Roy.  Microsc.  Soc.,  Ill 
(1880),  331-332. 

3  Eraile  Bertrand:  Nouveau  mineral  des  environs  de  Nantes.    Bull.  Soc.  Min.  France, 
III  (1880),  96-100. 

4  C.  Leiss:  Die  optischen  Instrumente  der  Firma  R.  Fuess.    Leipzig,  1899,  230-231. 

5  W.  C.  Brogger  und  Gust.    Flink:  Ueber  Krystalle  von    Beryllium  und    Vanadium. 
Zeitscher.  f.  Kryst.,  IX  (1884),  225-237,  in  particular  227-228. 

6  C.  Leiss:  Op.  a/.,  228-229. 


302 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  254 


microscope  by  the  spring  clips,  is  an  upright  i  through  which  passes  a  hori- 
zontal axis.  At  one  end  is  a  milled  head  k,  and  at  the  other  a  disk  v.  The 
amount  of  rotation  may  be  read  in  degrees  from  the  graduated  circle  T.  The 
disk  v  has  a  milled  edge  w,  and  may  be  rotated,  carrying  with  it  a  central 
hemisphere  h.  The  latter  is  loosely  placed  in  the  opening  of  the  disk  and  may 


FIG.  402. — Section  through  Brogger's  micro- 
goniometer. 


FIG.  403. — Brogger's   micro-goniometer. 
2/3  natural  size.      (Fuess.) 


be  tilted  at  any  angle,  being  retained  in  its  position  by  a  coating  of  some  heavy 
oil,  such  as  vaseline.  Through  the  center  of  the  hemisphere  is  bored  a  cone- 
shaped  hole,  larger  on  the  under  side  than  on  the  upper.  Projecting  into  its 
center,  and  lying  in  the  plane  of  the  flat  surface,  is  a  blunt  needle  N  which 
may  be  rotated  by  means  of  the  screw  S.  The  instrument,  being  intended 

primarily  for  adjusting  the  position  of  crystals 
to  read  their  interfacial  angles  either  by  the 
stage  vernier  or  by  an  ocular  goniometer 
(Figs.  393-394),  is  provided  with  no  gradua- 
tions except  the  single  circle  T. 

Nachet's1  tub  goniometer  resembles  that 
of  Bertrand,  but  instead  of  inserting  the  hori- 
zontal axis  through  the  side  of  the  vessel,  he 
attached  it  to  a  geared  wheel  within  it. 
Another  wheel,  above  the  rim,  transmitted  the 
rotation  to  the  graduated  circle. 

In  1891,  Klein2  described  the  first  of  his 
many  rotation  apparatus,  and  it  is  chiefly  to 
him  and  to  von  Fedorow  that  our  modern  in- 
struments are  due.  Klein's  earliest  apparatus,  as  constructed  by  Fuess,3  is 
shown  in  Fig.  404.  It  consists  of  a  circular  metal  plate,  in  the  center  of 
which  is  a  hole,  partially  surrounded  by  a  metal  collar.  Into  this  is  in- 
serted, and  held  by  the  clamp  K,  the  immersion  vessel,  of  which  two  sizes  are 

1  A.  Nachet:    Cwoe  goniometre.     Bull.  Soc.  Min.  France,  X  (1887),  186-187. 

2  C.     Klein:    Krystallographisch-optische     Untersuchungen.     Ueber     Construction     imd 
Verwendung  von  Drehapparaten   zur    optischen    Untersuchung   von    Krystallen  in  Mcdien 
ahnlicher  Brechbarkeit.     Sitzb.  Akad.  Wiss.  Berlin,    1891  (I),  435-444,  in  particular  435- 

437- 

3  C.  Leiss:  Op.  «"/.,  231. 


FIG.  404. — Klein's  small  rotation  ap 
paratus.     1/2  natural  size.     (Fuess.) 


ART.  254]  ROTATION  APPARATUS  303 

provided.  The  crystal  to  be  examined  is  attached  to  the  blunt  end  of  a 
horizontal  glass  rod,  at  whose  other  end  is  a  circle  graduated  to  degrees. 
H  is  a  spring  to  keep  the  rotating  axis  in  close  contact  with  the  shoulder 
and  thus  prevents  the  escape  of  the  immersion  fluid,  which  should  have  a 
refractive  index  as  nearly  as  possible  the  same  as  that  of  the  substance 
under  examination. 

In  the  same  year  appeared  von  Fedorow's1  first  description  of  his  "  Uni- 
versal tisch."  It  was  described  in  greater  detail  in  1893,  in  the  second  of  a 
long  series  of  articles2  on  "Universal  methods  in  mineralogy  and  petrog- 
raphy," and  two  types  were  illustrated.  Both  of  these  were  considerably 


FIG.  405. — The  von  Fedorow  small  model  universal  FIG.  406. — Object-glasses  for  von  Fedo- 

stage.      1/2  natural  size.     (Fuess.)  row's  small  universal  stage. 

changed  at  various  times  until,  at  present,  the  two  types  appear  as  shown  in 
Figs.  405  and  407. 

The  first,  or  small  model3  (Fig.  405),  may  be  attached  to  the  stage  of  even 
the  smallest  microscopes,  since  the  height  of  the  object  slip  above  the  base 
is  only  1 1  mm.  Two  vertical  supports  ssr  carry  the  horizontal  axis,  which  is 
rotated  by  the  milled  head  k,  the  amount  being  read  to  5  degrees  from  the 

1  E.  von  Fedorow:    Eine  neue  Methode  der  optischen  Untersuchung  von  Kry  stall  platten 
in  pardlelem  Lichte.     T.  M.  P.  M.,  XII  (1891),  505-509. 

2  (a)  E.  von  Fedorow:  Universal-  (Theodolith-}  Method  in  der  Miner  alogie  und  Petro- 
graphie.     I  Theil.     Zeitschr.  f.  Kryst.,  XXI  (1892-1893),  574-714. 

(b)  Idem:  Same  title,  //  Theil.    Ibidem,  XXII  (1893-1894),  229-268. 

(c)  Idem:  Die  einfachste  Form  des  Uniiersaltischchens.     Ibidem,  XXIV,  602-603. 

(d)  Idem:  Optische  Mittheilungen.     Nock  ein  Schritt  in  der  Anwendung  der  Univer- 
salmethode  zu  optischen  Studien.     Ibidem,  XXV  (1895-1896),  351-356. 

(e)  Idem:  UnhersalmethodeundFeldspathstudien.     I.  Melhodische  Verfahren.     Ibidem, 
XXVI  (1896-1897),  225-261.     In  particular  226-230,  241-242. 

(f)  Idem:  Same  general  title,  //.  Feldspathbestimmungen.     Ibidem,   XXVII   (1897- 
1898),  337-398- 

(g)  Idem:  Same  general  title.     777.  Die  Feldspdthe  des  Bogoslowsk'schen  Bergreviers. 
Ibidem,  XXIX  (1897-1898),  604-658. 

See  also  the  following: 

(h)  C.  Leiss:  Vervollstdndigte  neue  Form  des  E.  v.  Fedorow'schen  Uniiersaltischcs. 
Neues  Jahrb.,  1807  (II),  93-94. 

(i)  Idem:  Uniiersaltische  einfachster  Form  nach  E.  v.  Fedorow.  Neues  Jahrb.,  B.  B,, 
X  (1895-1896),  420-423. 

(j)  Idem:  Die  optischen  Instruments •,  etc.,  Leipzig,  1899,  233-236. 

(k)  Fred.  Eugene  Wright:  The  measurement  of  the  optic  axial  angle  of  minerals  in  the 
ihin  section.  Amer.  Jour.  Sci.,  XXIV  (1907),  317-369,  in  particular  343. 

3  See  references  c,  e,  i,  j.  above. 


304  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  254 

graduated  circle  T,  and  to  single  degrees  from  the  vernier  n.  A  screw  / 
clamps  the  axis  in  any  desired  position.  As  in  all  of  von  Fedorow's  stages, 
circular  object  slips,  20  mm.  in  diameter  and  i  mm.  in  thickness,  are  used. 
In  this  stage  they  are  made  to  do  double  duty.  A  rabbet  in  the  plate  IP 
receives  them  and  permits  a  rotation  in  azimuth.  The  round  object-glasses 
O  are  marked,  near  the  periphery,  as  shown  in  Fig.  406,  with  four  scratches, 
separated  by  90°,  and  distinguished  by  one,  two,  three,  or  no  dots.  They 
are  pressed  against  the  graduated  side  of  the  plate  P  by  means  of  a  spring, 
thus  permitting  more  accurate  reading  of  the  scale,  which  is  divided  into  5° 
spaces.  The  rotation  may  be  estimated  to  one  degree.  If  the  table  is  con- 
siderably tilted,  two  lenses,  less  than  hemispheres  by  the  thickness  of  the 
object-  and  cover-glasses,  and  with  an  index  of  refraction  of  1.7  to  1.8,  may 

be  attached  below  and  above 
by  means  of  a  drop  of  gly- 
cerine, to  increase  the  angle  of 
vision.1 

The  large  model  (Fig. 
407) 2  has  four  movements. 
The  base  may  be  clamped 
to  the  stage  of  the  micro- 
scope by  means  of  the  thumb 
screws  tt.  Two  uprights  /  sup- 

FIG.  407.— The  improved  von  Pedorow  large  universal      port,  and  a  SCrCW/clampS  the 

.  h°riz°ntal  axis,  the  amount  of 

whose  rotation  may  be  read  to 

five  minutes  on  the  circle  T  and  the  vernier  n.  The  disk  T\  may  be  rotated 
by  means  of  a  tangent  screw,3  read  to  five  minutes  at  n,  and  clamped  in 
position  by  the  screw  g.  The  tilting  stage  K  may  be  clamped  by  the 
screw  d,  and  the  amount  of  inclination  read  to  degrees  on  two  hinged 
graduated  segments  V  and  Vi,  suggested  by  von  Fedorow4  but  first 
constructed  by  Wright.5  The  inner  disk  5*  is  of  glass  and  rotates  with- 
in the  graduated  circle  K.6  It  may  be  read  to  degrees.  If  a  Hirschwald 
stage  is  used,  the  instrument  may  be  permanently  clamped  to  a  blank  sliding 
plate,  which  itself  may  be  held  securely  in  position  on  the  microscope  by  a 
single  thumb  screw. 

As  in  the  small  model,  two  rather  less  than  hemispheres  of  glass,  with  or 
without  holders,  may  be  placed  above  and  below  the  preparation  and  so 

1  See  page  353  of  reference  d,  and  page  229  of  c. 

2  See  references  e,  h,  and  j,  above. 

8  An   addition  suggested  by  Albert  Johannsen  in  an  instrument  purchased  from  Fuess 
for  the  U.  S.  Geological  Survey. 

4  See  reference  e. 

5  See  reference  k. 

6  See  reference  e. 


ART.  254J 


ROTA  TION  APPA RA  Tl'S 


305 


arranged  that  the  thin  section  forms  the  center  of  the  sphere.     This  is  on  the 
same  principle  as  those  first  used  by  Adams1  in  his  polariscope. 

In  1893,  Klein2  described  three  new  rotation  apparatus.  The  first 
(Fig.  408)  is  so  arranged  that  a  crystal  attached  to  the  rod  F  may  be  tilted 
and  rotated  in  any  desired  direction.  When  used  with  an  immersion  fluid, 
the  microscope  is  inclined  backward  to  a  horizontal  position,  the  rotation 


K 


FIG.    408. — Klein's   Universaldrehapparat. 
1/2  natural  size.     (Fuess.) 


FIG.  409. — Klein's  apparatus  for  the  examination 
of  gems.     1/2  natural  size.     (Fuess.) 


apparatus  being  held  firmly  to  the  stage  by  two  strong  clamps,  and  the  liquid 
is  placed  in  a  vessel  which  is  held  in  position  on  a  separate  support.  The 
instrument  has  three  rotation  motions,  one  around  a  complete  circle,  meas- 
ured by  graduations  on  K,  and  two  90°  rotations,  L  and  LI.  All  readings  may 
be  made  by  means  of  verniers  to  5  minutes.  P  is  a  rod  by  means  of  which  the 
mineral  attached  to  F  may  be  lowered,  and  V  and  S  are  clamping  screws. 

The  second  instrument  (Fig.  409),  with  two  motions  at  right  angles  to 
each  other,  was  designed  especially  for  the  examination  of  gems.  Like  the 
preceding  it  must  be  used  with  the  microscope  in  a  horizontal  position.  The 
containing  vessel  for  the  immersion  fluid  may  be  readily  clamped  on  or  re- 
moved. The  motion  P  is  of  90°,  that  of  K,  360°.  Verniers  on  each  make 
possible  readings  to  10  minutes.  In  addition,  by  means  of  the  screws  S  and 
T,  one  may  adjust  the  crystal  exactly  in  the  axis  of  the  microscope. 

1  W.  G.  Adams:  A  new  polariscope.     Phil.  Mag.,  L  (1875),  ^3~17- 

Abstract  of  same  article:  Ueber  ein  neues  Polariskop,  Pogg.  Ann.  CLVII  (1876),  297- 
302. 

2  C.  Klein:    Der  Uniiersaldrehapparat,  ein  Instrument  zur  Erleickterung  und  Vcrein- 
fachung   krystdlographisch-optisclier    Untersuchungm.     Sitzb.    Akad.    Wiss.    Berlin,    1895 
(I),  91-107. 

C.  Leiss:  Ueber  Neuconstructionen  ion  Instrumental  fiir  krystallographische  und  petro- 
graphische  Untersuchungen.     Neues  Jahrb.     B.  B.  X  (1895-6),  187-189. 
Idem:  Die  cptischen  Inslrumcnte,  etc.,  232-233. 
20 


306  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  254 

The  third  instrument1  (Fig.  410)  differs  somewhat  from  the  preceding 
in  being  designed  principally  for  the  examination,  in  parallel  light,  of  thin 
sections  immersed  in  a  highly  refracting  liquid.  The  tub  for  the  immersion 
fluid  is  large  enough  to  permit  the  examination  of  every  part  of  a  rock  section. 
15  mm.  square,  when  mounted  on  an  object  slide  28  by  48  mm.  The  micro- 
scope is  placed  in  a  vertical  position,  and  the  apparatus  is  attached  by  means 
of  two  strong  clamps.  In  the  bottom  of  a  hemispherical  tub  B  is  a  glass 
plate  a  for  the  transmission  of  the  light  from  below.  A  horizontal  disk  7\,  in 

the  center  of  which  there  is  a 
glass  plate  S,  may  be  rotated 
in  azimuth  by  means  of  the 
button  k,  which  transmits, 
through  the  horizontal  axis  z, 
a  motion  to  a  geared  wheel. 
The  latter  engages  in  a  circular 
rack  TI  around  the  periphery  of 
the  disk,  and  the  amount  of  rota- 
tion may  be  read,  to  five  min- 
utes, by  the  vernier  n\.  Two 

FIG.  410. — Klein's  tub  goniometer.      1/2  natural  size.  .  ,.  ,  111^1 

(Fuess-)  spring  clips,  e  and  el}  hold   the 

section  in   place.     A  movement 

in  altitude  is  produced  by  the  wheel  T,  and  may  be  read  to  five  minutes 
by  the  vernier  n,  c  being  the  axis  by  means  of  which  this  motion  is  pro- 
duced. In  a  later  paper,  Klein2  described  two  further  motions  which 
were  added  to  the  inner  disk  by  cutting  it  into  concentric  circles,  to  one  of 
which  was  imparted  a  tilting  motion  in  altitude,  and  to  the  other  a  hori- 
zontal rotary  motion,  the  latter  of  which  was  still  later3  graduated. 

In  1895  Schroeder  van  der  Kolk4  described  a  very  simple  device  by  means 
of  which  a  crystal  or  thin  section  might  be  tilted  for  examination  in  any 
direction.  It  consisted  simply  of  a  hemisphere  of  glass,  30  mm.  in  diameter, 
the  convex  side  of  which  rested  in  the  opening  of  the  stage  of  the  microscope. 
The  slide  was  placed  upon  the  flat  surface  and  was  held  in  place  by  a  drop  of 
glycerine  or  oil.  If  it  was  desired  to  place  the  hemisphere  so  that  the  upper 

1  C.  Klein:    Bin  Unhersaldrehapparat  zur  Untersuchung  von  Dunnschlifen  in  Fliissig- 
keiten.     Sitzb.  Akad.  Wiss.  Berlin,  1895  (II),  1151-1159. 

C.  Leiss:  Ueber  neuere  Instrumente  und  V orrichtungen  fur  petrographische  und  krystal- 
lographische  Untersuchungen.     Neues  Jahrb.  B.  B.,  X  (1895-6),  423-425. 
Idem:  Die  optischen  Instrumente,  etc.,  237-238. 

2  C.  Klein:    Ueber  Leucit  und  Analcim  und  ihre  gegenseitigen  Beziehungen.     Sitzb. 
Akad.  Wiss.  Berlin,  1897  (I),  290-354,  in  particular  328-330. 

Idem:  Same  title.     Neues  Jahrb.  B.B.,  XI  (1897),  475-553,  in  particular  522-525. 

3  Idem:    Die  Anwendung  der  Methode  der  Totalreflexion  in  der  Petrographie.     Sitzb. 
Akad.  Wiss.  Berlin,  1898  (i),  317-331,  in  particular  footnote  2,  page  321. 

4  J.  L.  C.  Schroeder  van  der  Kolk:    Zur  Systembestimmung  mikroskopischer  Krystalle. 
Zeitschr.  f.  wiss.  Mikrosk.,  XII  (1895),  188-92. 

Idem:  Kurze  Anleitung  zur  Mikroskopischen  Krystallbestimmung.  Wiesbaden,  1898, 
37-39- 


ART.  254]  ROTATION  APPARATUS  307 

surface  was  exactly  horizontal,  all  that  was  necessary  was  to  press  down  upon 
it,  very  gently,  with  the  front  lens  of  a  medium  power  objective.  No  means 
of  measuring  the  amount  of  tilting  was  possible. 

A  variation  of  the  above,  made  by  ten  Siethoff,1  permits  examinations 
to  be  made  by  convergent  as  well  as  by  parallel  light.  This  device  is  espe- 
cially convenient  for  the  examination  of  small  crystals,  which  may  be  fastened 
to  the  upper  surface  of  the  hemisphere  by  means  of  balsam  or  oil.  This  hemi- 
sphere forms  the  upper  member  of  a  triple- 
lens  condenser  (Fig.  411)  and  is  entirely 
detached  from  the  casing  so  that  it  may  be 
rotated  in  any  direction.  In  order  that  high 
power  objectives,  consequently  those  of  short 
focal  lengths,  may  be  used,  the  upper  edges 
are  beveled,  and  the  whole  instrument  is  of 
such  size  that  it  may  be  slipped  into  the 

central  opening  of  the  stage  of  the  micro-     PlG-  "™3  "" 

scope,  being  held  in  place  by  the  pressure 
of  the  spring  object  clips  on  the  collar  t. 

Another  variation  of  the  Schroeder  van  der  Kolk  instrument  is  that  of 
ArschinowT,2  who  surrounded  the  upper  edge  of  a  glass  hemisphere,  50  to  60 
mm.  in  diameter,  by  a  metal  band  to  which  were  pivoted,  at  right  angles  to 
each  other,  two  graduated  arcs  5  mm.  wide.  Upon  the  face  of  the  hemisphere 
two  lines  were  engraved,  joining  the  pivot  points  of  the  arcs.  An  ebonite 
ring,  cut  out  to  receive  the  glass  hemisphere  and  lined  with  chamois  skin,  is 
clamped  upon  the  stage  of  the  microscope  and  holds  the  device  in  place. 
The  thin  section  to  be  examined  is  fastened,  cover-glass  down,  to  the  surface 
of  the  instrument  with  glycerine  or  cedar  oil,  and  it  is  tilted  to  the  desired 
position.  If  it  is  necessary  to  incline  the  section  greatly,  a  small  plano- 
convex lens,  differing  from  a  hemisphere  by  the  thickness  of  the  object  glass, 
is  placed  above  the  object3  to  enlarge  the  field  of  view.  After  the  crystal  or 
thin  section  has  been  examined,  the  inclination  of  the  hemisphere  is  deter- 
mined by  raising  the  two  graduated  arcs  until  they  intersect  directly  under 
the  cross-hairs  of  the  microscope,  and  reading  the  values  there  indicated 
with  the  same  objective  as  that  with  which  the  examination  of  the  mineral 
was  made.  The  graduations  of  the  arcs  are  in  degrees  with  o°  at  the  center 
and  90°  at  the  binges. 

Wright4  was  able  to  determine,  within  a  degree,  the  amount  of  rotation 

1  E.  G.  A.  ten  Siethoff:    Beitrag  zur  Krystalluntersuchung  im  convergenten  polarisirten 
Lichte.     Centralbl.  f.  Min.,  etc.,  1903,  657-658. 

2  Wladimir  Arschinow:    Ueber  die    Verwendung   einer  Glashalbkugel  zu  quantitatiien 
optischen  Uniersuchungen  am  Polarisationsmikroskope.     Zeitschr.  f.  Kryst.,  XL VIII  (1910- 
n),  225-229. 

3  Cf.  footnote  23,  supra. 

4  Fred.  Eugene  Wright:    The  methods  of  petro graphic-microscopic  research.     Carnegie 
Publication  No.  158,  Washington,  1911,  175. 


308  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  254 

of  a  hemisphere  of  glass,  63  mm.  in  diameter,  by  having  it  engraved  with 
parallels  and  meridians,  5°  apart  (Fig.  412).  The  opening  of  the  stage  in 
which  the  hemisphere  rested  (Fig.  413)  coincided  exactly  with  the  20°  parallel, 
and  two  small  notches,  cut  in  the  edge  of  this  opening,  indicated  the  zero 
meridian.  Across  the  flat  surface  of  the  hemisphere  two  lines  were  en- 
graved, crossing  at  right  angles  at  the  center,  to  assist  in  centering. 

A  very  simple  rotating  instrument  was  suggested  by  Jaggar1  in  1897. 
It  contains  no  graduated  circles  and  was  designed  simply  to  tilt  sections 


FIG.    412.  FIG.  413. 

FIGS.  412  and  413. — Wright's  hemispherical  rotation  apparatus. 

into  position  to  obtain  maximum  extinction  angles,  maximum  interference 
colors,  or  to  change  slightly  the  orientation  of  interference  figures  in  such 
cases  where  a  measure  of  the  amount  of  the  rotation  is  not  required.  The 
instrument  is  attached  to  the  stage  by  means  of  pins  in  the  object-clip  holes. 
It  consists  of  a  pair  of  spring  clips,  supported  1 5  mm.  above  the  stage  by  a 
ball-and-socket  joint  which  may  be  moved  in  any  direction  by  means  of  a 
long  removable  key.  A  thumb-screw  controls  the  tension  on  the  ball-and- 
socket  joint  by  means  of  pressure  against  a  brass  plate  faced  with  cork  which 
fits  the  ball.  The  amount  of  rotation  possible  is  about  45°. 

Recently  a  rotation  microscope2  has  been  placed  upon  the  market,  an 
instrument  especially  convenient  since  thin  sections  of  the  usual  size  may 
be  used.  It  is  described  and  figured  in  Article  184. 

1  T.  A.  Jaggar,  Jr. :    A  simple  instrument  for  Inclining  a  preparation  in  the  microscope. 
Amer.  Jour.  Sci.,  Ill  (1897),  129-131. 

2  Cf.  also  the  microscope  described  in  Art.  183.     In  this,  however,  the  universal  stage 
is  of  extraordinary  size. 


CHAPTER  XIX 
THE  COLOR  OF  MINERALS 

255.  Idiochromatic  and  Allochromatic  Minerals. — Color1  is  another 
property  of  minerals  which  may  be  determined  by  ordinary  light,  and  all 
colored  minerals  may  be  divided  into  two  classes,  those  that  are  idiochromatic 
and  those  that  are  olio  chromatic.  In  the  first  the  color  is  due  to  a  property 
of  the  mineral  itself,  namely  its  power  to  absorb  light  of  certain  wave  lengths. 
This  property  of  absorption,  however,  may  not  be  the  same  in  every  direction, 
or  different  wave  lengths  of  light  may  be  absorbed  and,  in  consequence,  the 
mineral  may  show  what  is  known  as  dichroism  or  pleochroism.2  In  the  sec- 
ond, the  color  is  due  to  minute  inclusions.  The  latter  may  be  of  such  size 
that  they  can  be  distinguished  under  the  microscope,  or  they  may  be  so  small 
and  so  sparsely  distributed  that  they  cannot  be  seen  even  with  the  highest 
powers.  They  are  then  spoken  of  as  "dilute"3  colors. 

As  to  the  nature  of  the  coloring  material,  there  exists  great  diversity  of 
opinion.  Being  so  dilute,  attempts  to  analyze  it  years  ago  resulted  in  pro- 
ducing the  opinion  that  they  were  volatile  organic  substances.  Thus 
Schneider4  thought  the  color  of  gems  due  to  hydrocarbons. 

Among  the  advocates  of  the  organic  nature  of  the  coloring  material  were 
Wyrouboff,  who  experimented  on  fluorite,  and  Kraatz-Koschau  and  Wohler, 
who  determined  the  presence  of  carbon,  nitrogen,  and  hydrogen  in  zircon, 
smoky  quartz,  amethyst,  fluorite,  apatite,  calcite,  microcline,  baryte,  rock- 
salt,  and  topaz.  Among  those  who  insisted  on  the  inorganic  nature  of  the 
pigment  were  Becquerel  and  Moissan,  and  Loew,  who  found  free  fluorine 
in  fluorite;  Weinschenk,Lehmann,  Rosenbusch,  and  Spezia,5  who  found  traces 
of  iron  in  brown  zircon. 

There  seems  to  be  no  doubt  that  the  color  of  minerals  of  the  second  class 
is  always  produced  by  some  foreign  substance  although  its  nature  may  or 
may  not  be  known.  Without  a  doubt,  in  some  minerals,  it  is  inorganic; 
less  clearly  proven,  in  others,  is  its  organic  nature.  Whatever  the  pigment 
may  be,  it  is  distributed,  in  some  places  evenly,  in  others  irregularly,  through 
the  mineral,  but  often  so  sparingly  that  a  thin  section  appears  absolutely 

1  See  General  Bibliography  at  end  of  Chapter. 

2  See  Chap.  XXI. 

3  H.  Fischer:  Op.  cit.,  II  Abth.  See  General  Bibliography  at  end  of  Chapter. 

4J.  Schneider:  Ueber  Phosphor escenz  durch  mechanische  MitteL  Pogg.  Ann.,  XCVI 
(1855),  282-287. 

6  Giorg.  Spezia:  Sul  color e  del  zircon.  Atti  della  Reale  Accad.  delle  Scienze  di  Torino, 
XII  (1876).*  Review  in  Neues  Jahrb.,  1877,  303-305. 

Idem:  Same  title.  Atti  della  Accad.  etc.  Toririo,  XXXV  (1889).*  Review  in  Neues 
Jahrb.,  1900  (II),  344. 

309 


310  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  256 

colorless.  Not  only  may  the  coloring  matter  be  irregularly  distributed,  but 
two  colors  may  appear  in  the  same  crystal.  That  radium  has  some  effect  in 
bleaching  or  coloring  minerals  may  be  seen  by  the  pleochroic  halos  about 
certain  included  radioactive  minerals,  such  as  zircon  in  biotite  or  cordierite, 
but  that  radium  can  produce  a  color  in  a  naturally  colorless  mineral  is  not 
proven. 

Partly  owing  to  the  fact  that  the  color  of  minerals  is  not  a  constant 
property,  and  partly  because  no  simple  color  tables  have  been  available, 
colors  are  named,  at  the  present  time,  just  as  they  were  by  Werner,  over  a 
hundred  years  ago.1 

256.  Determination  of  Color. — Long  ago  Fischer2  attempted  to  classify 
definitely  the  color  of  minerals,  and  found  in  Radde's  color  scale3  a  means  of 
comparison.  This  work,  now  long  out  of  print,  consisted  of  a  series  of  colors 
arranged  in  the  order  of  the  spectrum.  The  main  divisions,  vermilion,  orange, 
yellow,  yellowish  green,  grass-green,  bluish  green,  blue,  violet,  purple,  and 
carmine,  graded  into  each  other  and  formed  30  transition  members.  In 
addition  to  these,  there  were  twelve  other  colors,  neutral  gray,  vermilion 
gray,  brown,  orange-gray,  yellowish  gray,  yellowish  greenish  gray,  greenish 
gray,  bluish  greenish  gray,  bluish  gray,  violet-gray,  purplish  gray,  carmine- 
gray.  At  right  angles  to  these  forty-two  colors  were  arranged  twenty-one 
tones  of  each,  ranging  from  black  to  nearly  colorless,  and  lettered  from  a  to  v. 
To  designate  any  particular  color  it  was  simply  necessary  to  use  a  number  and 
a  letter,  as  30^,  2i/,  etc. 

Owing  to  the  fact  that  Radde's  color  scale  is  now  almost  unobtainable, 
Moller4  proposed  Klincksieck  et  Valette's  Code  des  Couleurs  as  a  standard. 
There  has  recently  appeared  a  work  by  Ridgway5  in  which  1115  color  tints 
are  shown  arranged  by  tints  in  a  manner  similar  to  Radde's.  An  objection 
to  any  color  scale  of  this  kind  is  the  impossibility  of  matching  the  colors  seen 
under  the  microscope  by  transmitted  light  with  the  opaque  colors  of  the  scale 
seen  by  incident  light. 

Numerous  devices  have  been  suggested  for  producing  the  colors  of  the 
spectrum  for  comparison  with  the  transparent  colors  seen  under  the  micro- 
scope. While  such  instruments  are  of  considerable  value  in  the  comparison 
of  interference  colors,  and  as  such  will  be  described  below,  they  do  not  greatly 

1  The  list  of  colors  with  their  subdivisions  may  be  found  in  many  mineralogies,  for  ex- 
ample in: 

Gustav  Tschermak:  Lehrbuch  der  Mineralogie.     Wien,  3  Aufl.,  1888,  156-157. 
Wilhelm  Haidinger:  Handbuch  der  bestimmenden  Mineralogie.     Wien,  i  Aufl.,   1845, 
332-343. 

2  H.   Fischer:  Ueber  die  Beziechnung  von  Farbenabstufungen  bei  Miner  alien.     Neues 
Jahrb.,  1879,  854-857- 

3  International    Farbenskale   von    Radde   in    Hamburg.     Societe    stenochromique, 
Paris.* 

*  Hans  Jakob  Moller:  International  Farbenbestimmungen.  Ber.  deutsch.  pharmazeut. 
Gesell.,  1910,  358-368.  Review  in  Neues  Jahrb.,  1911  (II),  162. 

6  Robert  Ridgeway:  Color  standards  and  nomenclature.  Pp.  iii+44,  pi.  53-  Washing- 
ton, D.  C.,  1913.*  Review  by  W.  J.  Spillman,  Science,  XXXVII  (1913),  985-989- 


ART.  257]  THE  COLOR  OF  MINERALS  311 

help  in  the  determination  of  the  ordinary  colors  of  minerals,  the  latter  not 
being  the  pure  colors  of  the  spectrum. 

This  method  of  comparison  was  suggested,  in  1849,  by  Briicke,1  who 
proposed  a  gypsum  wedge,  mounted  between  glass  plates,  as  a  means  of 
producing  the  colors.  Later2  he  invented  a  simple  apparatus  which  he  called 
a  "Schistoskop,"  in  which  interference  colors,  produced  by  gypsum  plates, 
were  used  for  comparison.  Arons,3  in  1910,  described  a  "  chromoscope " 
in  which  the  colors  are  produced  by  the  passage  of  light,  between  crossed 
nicols,  through  quartz  plates  of  different  known  thicknesses  and  cut  at  right 
angles  to  the  axis,  the  variation  being  produced  by  rotating  the  nicol  through 
some  angle  less  than  180°.  A  quartz  wedge  may  be  used  instead  of  quartz 
plates  of  different  thicknesses.  Wright4  suggested  that  the  Ives  colorimeter 
might  be  used  with  the  microscope.  This  consists  of  red,  green,  and  blue  ray 
niters  so  arranged  that  when  the  three  are  simultaneously  viewed,  the  light 
is  white.  The  screens  are  mounted  on  a  disk  driven  rapidly  by  an  electric 
motor,  and  the  amount  of  each  light  is  regulated  by  shutters  so  made 
that  the  percentage  of  each,  used  in  producing  the  proper  color,  can  be 
determined. 

Nutting5  described  and  illustrated  an  apparatus  in  which  the  spectral 
colors  are  used  for  comparison;  the  various  shades  being  produced  by  the 
admission  of  more  or  less  white  light. 

257.  Determination  of  the  Color  of  Opaque  Minerals. — For  the  observa- 
tion of  the  colors  of  opaque  minerals,  an  apparatus  was  invented  by  Inostran- 
zeff.6  As  originally  made,  the  color  of  the  mineral,  viewed  directly  through 
the  microscope,  was  compared  with  a  standard  mineral  by  means  of  reflection 
through  two  prisms  from  a  known  mineral  in  another  microscope.  The 
device  was  not  satisfactory,  however,  since  the  comparison  was  made  between 
a  color  seen  directly  and  one  seen  only  by  reflection.  Inostranzeff,  therefore, 
improved  the  double  ocular  by  placing  the  eyepiece  intermediate  between  the 
two  microscopes  (Fig.  414),  from  each  of  which  appears  one-half  the  field,  one 
with  the  unknown  and  one  with  the  known  mineral  for  comparison.  If  two 
opaque  minerals  of  the  same  kind  are  brought  to  the  center  of  the  field,  no 
separating  line  will  be  seen  between  them.  If  there  is  even  a  very  slight  differ- 
ence, the  line  will  appear.  The  scale  for  comparison,  instead  of  being  made  up 
of  the  natural  minerals,  which  would  be  very  expensive,  is  prepared  from  the 
powder  of  the  minerals,  and  reproduces  both  color  and  luster  very  well. 

1  Ernst  Briicke:    Ueber  die  Aufeinanderfolge  der  Farben  in  den  Newton' schen  Ringen. 
Pogg.  Ann.,  LXXIV  (1849),  582-586. 

2  Idem:    Die  Physiologic  der  Farben  fur  die  Zwecke  der  Kunstgewerbe.     Leipzig,  1887. 

3  Leo  Arons:    Ein  Chromoskop.     Ann.  d.  Phys.,  4  ser.,  XXXIII  (1910),  799-832. 

4  Fred.  Eugene  Wright:    The  methods  of  petrographic-microscopic  research.     Carnegie 
Publication  No.  158,  Washington,  1911,  69. 

6  P.  G.  Nutting:    Outline  of  applied  optics.     Philadelphia,  1912,  Chapter  VI. 
6  A.    v.   Inostranzeff:    Ueber   eine    Vergleichungskammer   zur   mikroskopischen    Unter- 
suchung  undurchsichtiger  Mineralien.     Neues  Jahrb.,  1885  (II),  94-96. 


312  MANUAL  OF  PETROGRAPHIC  METH&DS  [ART.  257 

A  modification  of  this  apparatus,  giving  a  field  divided  horizontally  into 
halves  (Fig.  415),  was  made  by  Van  Heurck1  for  comparing  diatoms.  This 
suggests  the  use  of  such  a  device  for  comparing  thin  sections  of  rocks  from  any 
region,  or  sections  of  similar  rocks  from  different  regions. 


4 — p--{v 


o 

PIG.  414. — Inostranzeff's  comparateur.  FIG.  415. — Van  Heurck's  comparateur. 

GENERAL  BIBLIOGRAPHY 

1862.  H.  Rose:  Ueber  blaues  Steinsalz.  Zeitschr.  d.  deutsch.  geol.  Gesell.,  XIV  (1862),  4-5. 
1866.  E.  Reichert:  Das  Steinsalzbergwerk  Stassfurt  und  die  Vorkommnisse  in  demselben. 

Neues  Jahrb.,  1866,  321-350. 
G.   Wyrouboff:    Sur    les    substances    color  antes    des  fluorines.     Bull.  Soc.  Chim. 

Paris,  V  (1866),  334-347- 
1871.  A.  Forster:  Studien  uber  die  Farbung  der  Rauchquarze  oder  sogenannten  Rauchtopase. 

Pogg.  Ann.,  CXLIII  (1871),  173-194. 
1881.  O.  Low:  Freies  Fluor  im  Flussspath  von  Wb'lsendorf.     Ber.  deutsch.  Chem.  Gesell., 

XIV  (1881),  1144-1146. 
1885.  Edm.  Becquerel:  Etude  spectrale  des  corps  rendus  phosphor escents  par  Inaction  de  la 

lumiere  ou  par  les  decharges  electriques.     Comptes  Rendus,  CI  (1885),  205-210. 
H.  Fisher:  Kritischen  mikroskopisch-mineralogischen  Studien.* 

1890.  Henri  Becquerel  et  Henri  Moissan:  Etude  de  la  fluorine  de  Quincie.     Comptes  Ren- 

dus, CXI  (1890),  669-672. 

1891.  O.  Lehmann:  Ueber  kiinstliche  Farbung  ion  Krystallen.     Zeitschr.  f.  phys.  Chemie., 

VIII  (1891),  543-553- 
1896.  A.  Pelikan:  Ueber  den  Schichtenbau  der  Krystalle.     T.  M.  P.  M.,  XVI  (1896-7), 

1-64,  in  particular  46-50. 

E.  Weinschenk:  Die  Farbung  der  Mineralien.     Zeitschr.  d.  deutsch.  geol.  Gesell., 
XLVIII  (1896),  704-712. 

1898.  K.  v.  Kraatz-Koschlau  und  Lothar  Wohler:  Die  naturlichen  Farbungen  der  Minera- 

lien.   T.  M.  P.  M.,  XVIII  (1898-99),  304-333,  447-468. 

1899.  E.  Weinschenk:  Natiirliche  Farbung  der  Mineralien.     T.  M.  P.  M.,  XIX  (1899-1900), 

144-147. 
Joh.  Koenigsberger:  Ueber  die  farbende  Substanz  im  Rauchquarz.   T.  M.  P.  M.,  XIX 

(1899-1900),  148-154. 
Arnold  Nabl:  Ueber  farbende  Bestandtheile  des  Amethysten,  Citrins  und  gebrannten 

Amethysten.     Sitzb.  Akad.  Wiss.  Wien.,  CVIII  (1899),  Abth.  II,  48-57. 

1900.  Arnold  Nabl:  Natiirliche  Farbung  der  Mineralien.     T.  M.  P.  M.,  XIX  (1899-1900), 

273-276.  ^ 

1903.  Carl  Ochsenius:  Blaues  Steinsalz.     Centralbl.  f.  Min.  etc.,  1903,  381-383. 

1904.  Hans  Dudenhausen:  Optische  Untersuchungen  an  Flussspath  und  Steinsalz.     Neues 

Jahrb.,  1904  (I),  8-29. 

1906.  E.  Wulfing:  Einiges  uber  Miner alpigmente.     Festschrift  Harry  Rosenbusch.     Stutt- 
gart, 1906,  49-67. 

Fr.  Focke  und  Jos.  Bruckmoser:  Ein  Beitrag  zur  Kenntniss  des  blaugefdrbten  Stein- 
salzes.    T.  M.  P.  M.,  XXV  (1906),  43-60. 

1908.  K.  Simon:  Beitrdge  zur  Kenntniss  der  Mineralfarben.  Neues  Jahrb.  B.  B.,  XXVI 
(1908),  249-295. 

1910.  C.  Doelter:  Das  Radium  und  die  Farben.     Dresden,  1910,  133  pp. 

1911.  R.  Brauns:  Die  Ursachen  der  Farbung  dilut  gefdrbter  Mineralien  und  die  Einfluss  von 

Radiumstrahlen  auf  die  Farbung.     Fortschritte  der  Min.,  Kryst.,  und  Petrog,  I 
(1911),  129-140. 

1  Van  Heurck:    Bull.  Soc.  Belg.  Microsc.,  XIII  (1886),  76-78.*     Review  Van  Heurr.k's 
comparator.     Jour.  Roy.  Microsc.  Soc.,  1887,  463-464. 


CHAPTER  XX 


MONOCHROMATIC  LIGHT 

258.  The  Production  of  Monochromatic  Light. — For  most  petrographical 
determinations,  ordinary  white  light  answers  the  purpose,  but  for  very  exact 
measurements  it  is  necessary  to  use  monochromatic  light.  As  an  example, 
the  case  of  refractive  indices  may  be  cited.  From  the  well-known  phenome- 
non of  the  spectrum,  it  may  be  seen  that  light,  in  passing  through  a  prism, 
is  broken  up  into  rays  having  greater  or  less  angles  of  refraction  (Fig.  78). 
But  with  an  increased  angle  of  refraction  there  is  also  an  increased  value 
for  the  sine,  and  with  this  a  decreased  value  in  the  refractive  index.  The 
consequence  is  that  white  light,  passing  from  a  rarer  to  a  denser  medium, 
emerges  with  rays  having  different  refractive  indices,  that  of  the  violet,  which 
is  bent  most  from  its  original  course  and  has  the  least  angle  of  refraction,  is 
the  highest,  and  that  of  the  red,  which  is  bent  least,  is  the  lowest.  Thus  in 
crown  glass  the  index  for  the  A  line  (red)  is  1.5089,  for  D  (yellow)  1.5146, 
and  for  H  (violet)  1.5314. 

TABLE  OP  WAVE  LENGTHS  1 


Color 

Fraunhofer 
line 

Wave  length  X 

Produced 
by 

Red.. 

769  .93/^1" 

'i  

766.56 
759-40 
686  .  74 
670.82 

K 

Li 

Orange  

c 

656.30 
610.38 

H 

Li 

Yellow 

D, 

c;8o  62 

Na 

Green  .  . 

D\ 

589.02 
535  .06 

Na 
Tl 

£1 

527  .05 

E-2 

526  .97 

bi 

ei8    38 

Blue   ... 

F 

486    15 

H 

f  . 

434  oo 

Indigo  
Violet  

G 

h 
H 

430  •  79 
410.  19 
306  8  1 

"H" 

1  Louis  Bell:  The  absolute  wave  length  of  light.     Phil.  Mag.,  XXV  (1888),  245-263, 
350-372. 

Henry  A.  Rowland:  A  new  table  of  standard  wave  lengths.     Astron.  and  Astrophys., 
XII  (1893),  321-347- 

313 


314 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  259 


The  usual  methods  of  producing  monochromatic  light  are  as  follows: 

1.  By  means  of  ray  filters. 

2.  By  the  vaporization  of  certain  solids. 

3.  By  means  of  incandescent  gases. 

4.  By  means  of  certain  rays  of  the  spectrum,  separated  by  a  mono- 
chromator. 

259.  Ray  Filters. — Truly  monochromatic  light  cannot  be  produced  by 
the  absorption  of  the  other  colors  by  means  of  ray  filters,  although  such  de- 
vices suffice  for  many  purposes.  Thus  a  glass  coated  with  a  thin  film  of 
copper  oxide  will  transmit  light  between  the  Fraunhofer  lines  a  and  D; 
blue  cobalt  glass  will  permit  blue  and  extreme  red  to  pass.  More  satisfactory 
are  ray  filters  made  of  colored  solutions  enclosed  in  parallel-walled  glass  ves- 
sels, 15  to  20  mm.  between  walls,  and  used  in  combinations  to  give  any 
desired  color.  Landolt1  gives  the  following: 


Thick- 

Grams 

ness 

TV/T 

Color 

of 
filter 

Aqueous  solution  of 

per 

100  C.C. 

water 

^  in  /x/x 

/I 

mm. 

Red 

20 

Crystallized  violet,  5  BO  

o  .  oo  =5  \ 

20 

Potassium  chromate 

r 

IO    O 

639-718 

665.9 

Yellow  .  . 

20 

Sulphate  of  nickel  (NiSO4  +  7  aq) 

{ 
30.0 

15 

Potassium  chromate  

0 
10.  0         > 

574-614 

SQi-Q 

I  r 

Potassium  permanganate  

OO    O2^j 

Green  

2O 

Copper  chloride  (CuCl2  +  2  aq)  

60.0      1 

20 

Potassium  chromate  

10.  0         / 

505-540 

533-0 

Blue  (light).. 

2O 

Double  green  SF  

o.  02    \ 

green  494-526 

2O 

Copper  sulphate  (CuSO4+5  aq).  . 

15-0    J 

blue    458-494 

488.5 

Blue  (dark)... 

20 
20 

Crystallized  violet  5  BO  
Copper  sulphate  (CuSO4+5  aq).  . 

0.005  1 

15-0    J 

410-478 

448.2 

Crystallized  violet  $60  is  the  trade  name  for  the  chlorhydrate  of  hex- 
methyl  pararosaniline.  Double  green  SF  is  chlormethyl  hexmethyl  para- 
rosaniline  chlorhydrate  with  zinc  chloride.  The  pale  blue  light  is  not 
satisfactory  since  the  band,  from  458  to  526^,  is  too  broad  and  includes 
green  and  blue.  All  of  the  solutions  are  in  water  alone  except  the  crystallized 
violet  5BO,  the  crystals  of  which  should  be  dissolved  in  a  small  quantity  of 
alcohol  and  then  diluted  with  water  to  one  liter.  The  stock  of  the  two  aniline 
solutions  should  be  kept  in  the  dark,  the  others  do  not  alter  except  the  potas- 
sium permanganate  which  must  be  made  up  fresh  frequently. 

For  some  purposes,  one  or  more  2O-mm.  cells  cannot  be  used,  either  on 
account  of  their  thickness  or  on  account  of  the  reduction  of  light  by  reflection 

1  H.  Landolt:  Melhode  zur  Bestimmung  der  Rotations  dispersion  mil  Hiilfe  von  Strahlen- 
fillern.     Ber.  d.  d.  chem.  Ges.,  XXVII  (1894),  2872-2887,  in  particular  2884. 
Idem:  Sitzb.  Akad.  Wiss.  Berlin,  1894,  923. 
Idem:  Das  oplische  Drehungs-oermogen.     Braunschweig,  2  Aufl.,  1898,  387-390. 


ART.  259]  MONOCHROMATIC  LIGHT  315 

from  the  many  glass  cell- walls.  Nagel1  gives  a  list  of  fluids  which  can  be  used 
in  single  cells,  and  which  need  not  be  more  than  i  cm.  in  thickness.  The  mate- 
rials are  common,  the  solutions  are  easily  made,  and  will  keep  for  weeks  with- 
out precipitation  in  closed  cells.  No  proportions  are  given,  the  spectroscope 
being  used  to  determine  the  proper  light  transmitted.  The  solutions  are  as 
follows: 

Red:  Lithium-carmine  such  as  is  used  for  microscopic  coloring  material.  A 
thickness  of  i  mm.  gives  a  pure  red,  1/2  mm.  red  with  a  tinge  of  orange. 

Orange:  No  single  fluid  known  which  transmits  only  orange.  Aniline  orange 
lets  the  red  rays  pass  through;  a  solution  of  potassium  bichromate  i  cm.  thick  passes 
the  red,  orange,  yellow,  and  yellowish  green.  A  monochromatic  orange  filter  may 
be  made  by  preparing  a  not  quite  saturated  solution  of  copper  acetate  acidified  with 
a  few  drops  of  acetic  acid.  Add  slowly,  drop  by  drop,  enough  strong  saffranin  solu- 
tion to  extinguish  the  pure  yellow,  as  shown  by  the  spectroscope.  With  a  thick- 
ness of  i  cm.  the  visible  line  will  begin  near  the  C  line  and  end  with  the  D,  the  pure 
orange,  with  a  wave  length  of  640-6001*1*,  being  the  only  bright  color  transmitted. 

Yellow:  Pure  yellow  is  a  difficult  color  to  obtain  since  the  band  is  so  narrow.  A 
single  cell  i  cm.  thick  which  permits  rays  having  a  wave  length  between  620-570^ 
to  pass,  that  is  orange-yellow,  yellow,  and  greenish  yellow,  may  be  made  by  add- 
ing a  saturated  aqueous  solution  of  orange  G  to  an  acidified  copper  acetate  solution. 
The  solution  is  of  a  brown  color  and  does  not  keep  well. 

Greenish  yellow  and  yellowish  green:  A  very  transparent  filter  may  be  made  by 
boiling  an  excess  of  crystals  of  copper  acetate  in  a  saturated  solution  of  potassium 
bichromate  which  has  been  acidified  with  acetic  acid.  The  solution  should  be 
filtered  after  cooling.  The  58o-53o/x/z  waves  will  pass  through  a  cell  i  cm.  thick. 

Green:  If  one  dissolves  as  much  copper  acetate  as  possible  in  a  non-saturated 
solution  of  potassium  bichromate  or  picric  acid,  one  may  obtain  green  filters.  In- 
creasing the  amounts  of  the  potassium  bichromate  or  picric  acid  used  cuts  off  more 
and  more  from  the  blue-green  end. 

Pure  green  or  yellow-green:  To  a  saturated  solution  of  copper  ammonium  sul- 
phate with  an  excess  of  ammonia,  add,  drop  by  drop,  a  saturated  solution  of  potas- 
sium chromate,  until  the  entire  red,  orange,  yellow,  and  yellow-green  rays  are 
extinguished.  The  rays  transmitted  through  a  filter  0.7  mm.  in  thickness  have  a 
wave  length  of  535  to  4Q5MM-  The  blue-green  rays  are  removed  (535-510/1/1  trans- 
mitted) by  adding  to  the  above  a  few  drops  of  a  weak  alkaline  aqueous  solution  of 
fluorescine. 

Blue-green  and  cyan-blue:  In  an  acidified  copper  acetate  solution  drop  strong 
methyl  green  solution.  500  to  460/1/1  transmitted. 

Cyan  blue:  A  few  drops  of  gentian  violet  solution  added  to  the  preceding  makes 
a  pure  and  strong  blue.  Transmitted  rays  460-4501*1*. 

Blue  and  violet:  470-410^  may  be  cut  out  by  copper  ammonium  sulphate 
solution,  and  blue  and  violet  will  be  transmitted.  By  passing  the  rays  through 
another  cell  containing  a  dilute  solution  of  potassium  permanganate,  pure  violet 
results. 

MVilibald  A.  Nagel:  Ueber  fliissige  Strahlenfilter .  Biol.  Centralbl.,  XVIII  (1898), 
649-655. 


316  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  260 

Colored  gelatine  plates  have  been  used  as  ray  filters  by  Kirschmann1 
and  others,2  but  they  are  not  so  satisfactory  as  liquid  films,  since  they  are  not 
completely  transparent,  can  stand  neither  heat  nor  moisture,  and  are  not 
permanent,  the  aniline  color  fading.  They  are,  however,  usually  more  con- 
venient to  use  than  liquid  filled  cells.  The  simplest  way  to  prepare  such  a 
filter  is  to  fix  an  unexposed  dry  plate  in  hyposulphite  of  soda,  as  if  it  were  a 
negative,  then  place  the  gelatine-coated  plate  in  the  desired  color  and  dry 
in  a  dust-proof  room. 

For  yellow  use  i  grm.  Mars  yellow  in  200  c.c.  of  70  per  cent,  alcohol,  or  a  satu- 
rated solution  of  aurantia  in  alcohol.  For  red  dissolve  (a)  2  grm.  aurantia  in  40 
c.c.  absolute  alcohol,  (b)  5  grm.  rose  Bengal  in  20  c.c.  methyl  alcohol.  Mix  20  c.c. 
of  (a)  with  10  c.c.  of  (b)  and  90  c.c.  of  4  per  cent,  collodion.  For  green  use  copper 
nitrate  160  grm.,  chromic  acid  14  grm.,  distilled  water  250  grm.  Another  green 
is  eosine  or  malachite  green.  For  blue  use  methylene  blue. 

If  the  color  does  not  stain  the  gelatine  well,  it  should  first  be  mixed  with  4  per 
cent,  collodion. 

260.  Incandescent  Vapors  of  Solids. — A  limited  number  of  colors  may 
be  produced  by  the  vaporization  of  certain  solids.  The  method  is  very  simple 
and  the  colors  are  essentially  monochromatic.  The  salts  ordinarily  used  are 
lithium  sulphate  for  red  (X  =  670juju),  sodium  sulphate,  sodium  chloride,  or 
sodium  carbonate  for  yellow  (\=  589/1^),  and  thallium  sulphate  for  green 
(^=53 5 MM).  Yellow  light  is  the  one  most  commonly  used.  The  sodium 
chloride  gives  the  most  intense  light  but  the  carbonate  lasts  longer. 

For  any  of  these  colors  the  salt  may  be  enclosed  in  a  coil  of  platinum  wire, 
or  a  piece  of  pumice  may  be  saturated  with  a  solution,  and  placed  over  a 
Bunsen  or  alcohol  burner,  or  any  one  of  the  many  more  or  less  handy  devices 
for  the  vaporization  of  the  salt  may  be  used.3  Fig.  416  shows  a  very  handy 

1  A.  Kirschmann:  Ueber  die  Herstellung  monochromalischen  Lichtes.     Philos.  Studien 
von  W.  Wundt.  Bd.  VI  (1891),  543-552.* 

2  J.  William  Gifford:  An  inexpensive  screen  for  monochromatic  light.     Jour.  Roy.  Microsc. 
Soc.,  1894,  164-167. 

K.  Diederichs:  Die  Herstellung  von  gegossenen  Gelatineplatlen  ah  Strahlenfiller.  Zeit- 
schr.  f.  angew.  Mikrosk.,  IX  (1903),  197-198. 

Ernst  Pringsheim,  jim. :  Ueber  die  Herstdlung  von  Gelbfiltern  und  ihre  Verwendung  zu 
Versuchen  mil  lichtreizbaren  Organismen.  Ber.  deutsch.  Bot.  Gesell.,  XXVI  A  (1908), 
556-565- 

J.  Jullien:  Bull.  Spc.  Zoo),  de  Geneve,  1908,  104.*  Review  Economical  monochromatic 
filters.  Jour.  Roy.  Microsc.  Soc.,  1909,  522. 

3  See  H.  Landolt:    Das  optische  Drehungs-oermbgen.     Braunschweig,  2  Aufl,  1898,  353- 
359- 

Dr.  Pribram:  Ueber  einen  neuen  Brenner  fiir  Nalriumlicht.     Zeitschr.  f.  analytische 

Chemie,  XXXIV  (1895),  166. 

H.  Landolt:  Natriumlampe  fiir  Polarisationapparate.     Zeitschr.  f.  Instrum.,  IV  (1884), 

SPO- 
IL E.  J.  G.  du  Bois:  Ein  Intensivnatronbrenner.     Zeitschr.  f.  Instrum.,  XII  (1892), 

165-167. 


ART.  262] 


MONOCHROMATIC  LIGHT 


317 


burner.  It  consists  of  a  telescopic  Bunsen  burner,  which  may  be  raised  or 
lowered,  and  a  metal  chimney  to  preserve  a  steady  flame.  The  salt  is  placed 
in  a  platinum  cup  so  arranged  on  a  rod  that  it  may  be  instantly  thrown  in  or 
out  of  the  flame  by  means  of  the  pivot  c. 

Another  burner,  arranged  for  three  different  salts,  each  in  a  platinum  or 

asbestos  cup,  is  shown  in  Fig.  417.  The 
change  may  be  made  quickly  from  one 
colored  flame  to  another. 

A  number  of  very  elaborate  devices 
on  the  principle  of  an  atomizer,  are  given 
by  Beckmann.1 

It  is  very  desirable  that  a  hood  to 
carry  off  fumes  be  arranged  above  any 
burner  producing  monochromatic  light; 
thallium  fumes  because  they  are  poison- 
ous, and  sodium  because  the  minute  par- 
ticles will  long  remain  suspended  in  the 
air  and  will  overpower,  for  hours  after- 
ward, any  other  flame  that  may  be  used. 


n.Gr. 


FIG.  416. — Burner  for  producing  monochro-         FIG.  417- — Burner  for  producing  three  different 
matic  light.     1/5  natural  size.     (Fuess.)  monochromatic  lights.     (Steeg  und  Reuter.) 

261.  Incandescent  Gases. — Electricity  from  an  induction  coil,  passed 
through  a  Geissler  tube  filled  with  hydrogen,  will  give  a  spectrum  of  four  lines 
only,  namely  X  =  656.3/1/4,  486. i^t/*,  434.0/1/1,  and  410.2/6/4,  corresponding  to 
the  Fraunhof er  lines  C,  F,  f,  and  h.     By  the  use  of  suitable  ray  filters,  any  one 
of  these  lines  may  be  separated  from  the  others,  but  the  light  obtained  is 
not  very  intense. 

262.  Dispersed  White  Light    Produced  by  a   Monochromator. — The 

purest  monochromatic  light  that  can  be  produced  is  that  derived  from  the 
dispersion  of  white  light.     It  is,  however,  not  much  used  in  ordinary  petro- 

1  Ernst  Beckmann:  Ueber  Spektrallampen.     Zeitschr.  f.  phys.  Chemie,  XXXIV  (1900), 
593-611;  XXXV  (1900),  443-458,  652-660. 


318 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  262 


graphic  research  because  it  is  necessary  to  use  an  elaborate  and  expensive 
piece  of  apparatus.  Where  such  an  instrument  is  at  hand  for  mineralogical 
work,  it  may  well  be  used,  also,  with  the  microscope.  This  instrument  breaks 


E 


FIG.  418. — Wiilfing's  monochromator.     1/8  natural  size.     (Fuess.) 


up  white  light  into  a  continuous  spectrum  but  permits  only  the  desired  rays 
to  pass  out  through  a  narrow  slit,  perhaps  a  tenth  of  a  millimeter  in  width. 


FIG.  419. — Monochromator.     1/6  natural  size.     (Fuess.) 

An  instrument  of  this  kind,  really  a  spectroscope  with  the  addition  of  an 
adjustable  slit,  was  described  by  Tutton.1  The  source  of  light  is  an  electric 

1  A.  E.  Tuttcn:  An  instrument  of  precision  for  producing  monochromatic  light  of  any 
desired  wave-length,  and  its  use  in  the  investigation  of  the  optical  properties  of  crystals.  Phil. 
Trans.  Roy.  Soc.  London,  (A),  CLXXXV  (1894),  913-941. 

Idem:  Same  title  in  German.  Zeitschr.  f.  Kryst.,  XXIV  (1894-5),  455-474. 


ART.  262]  MONOCHROMATIC  LIGHT  319 

arc,  the  prism  is  one  of  60°,  perfectly  colorless,  with  refracting  faces  4  1/2  by 
2  1/2  in.  The  diffusion  of  the  light  is  produced  by  ground-glass  screens  of 
two  degrees  of  fineness.  Wiilfing1  described  a  similar  instrument  (Fig.  418) 
in  which,  however,  two  prisms  are  used  to  produce  the  spectrum  and  a  lens 
to  diffuse  the  light.  This  instrument  possesses  the  advantage  that  neither 
light  source  nor  examining  instrument  need  be  moved  to  change  from  one 
color  of  light  to  another,  the  change  being  produced  by  means  of  a  rotation 
of  the  prisms.  Daylight  or  electric  light  may  be  used.  In  the  Fuess2 
monochromator  (Fig.  419),  the  prism  used  is  a  Pellin  and  Broca  constant 
deviation  prism  with  a  dispersion  of  3°.  The  wave  length  of  the  light  emitted 
may  be  read  directly  from  the  large  drum  S,  graduated  from  390.0^  to 
,  which  controls  the  rotation  of  the  prism. 


1  E.  A.   Wiilfing:  Ueber  einen  Spectralapparat  zur  Herstellung  von  intensivem  mono- 
chromatischem  Licht.     Neues  Jahrb.  B.  B.,  XII  (1898-9)  343-404. 

2  C.  Leiss.    Zwei  Speklralapparate  (Monochromator  en)  zur  Beleuchtung  mil  homogenem 
Licht.     Zeitschr.  f.  Instrum.,  XXIX  (1909),  68-72. 


CHAPTER  XXI 
EXAMINATION  BY  PLANE  POLARIZED  LIGHT 

ABSORPTION,  DICHROISM,  PLEOCHROISM 

263.  Absorption  of  Light  in  Crystals. — Upon  its  emergence  from  any  sub- 
stance, the  intensity  of  light  is  more  or  less  reduced  from  that  with  which  it 
entered.     That  is  to  say,  a  certain  amount  of  light,  in  the  course  of  its  trans- 
mission, is  absorbed  by  the  body  through  which  it  travels.     If  this  absorption 
is  very  slight  and  the  amount  is  the  same  for  rays  of  every  wave  length,  the 
body  is  said  to  be  transparent  and  colorless.     If  the  absorption  of  certain  rays 
is  greater  than  others,  the  body  is  colored.     If  the  absorption  is  so  great  that 
even  in  very  thin  sections  no  light  passes  through,  the  body  is  opaque. 

If  we  construct  geometrical  solids  to  represent  the  amount  of  light  ab- 
sorbed after  passing  through  crystals  in  every  direction,  we  will  obtain 
figures  resembling  the  indicatrices.  These  figures  are  called  absorption  sur- 
faces and  differ  for  different  crystal  systems. 

264.  Isotropic  Substances. — Since  light  travels  with  equal  ease  in  every 
direction  in  isotropic  substances,  the  absorption,  for  any  color,  must  necessarily 
likewise  be  the  same  in  every  direction,  whereby  the  absorption  surface  will 
be  a  sphere,  and  all  sections  of  the  same  thickness  cut  from  a  mineral  will 
appear  of  the  same  shade  and  color.     For  any  other  wave  length  of  light, 
the  absorption  surface  is  a  sphere  whose  diameter  differs  from  the  first. 

265.  Anisotropic  Substances. — In  anisotropic  minerals,  absorption  may 
differ  in  different  directions,  whereby  sections  of  a  crystal,  cut  in  different 
directions  but  of  the  same  thickness,  may  appear  of  different  colors,  as,  for 
example,    cordierite   or   tourmaline.     This   property   of   crystals   is   called 
pleochroism  (or  dichroism1),  and  is  possessed,  to  a  certain  extent,  by  a  great 
many  minerals.     It  is  a  valuable  means  of  diagnosis,  and  may  be  determined 
very  simply,  under  the  microscope,  by  inserting  the  nicol  below  the  thin 
section  so  as  to  produce  plane  polarized  light,  that  is,  light  which  vibrates 
parallel  to  one  plane  only.     If  the  analyzer  were  inserted  instead  of  the  polar- 
izer, the  phenomenon  would  be  obscured  by  the  partial  polarization  produced 
by  reflection  from  the  mirror  below  the  mineral.     The  reason  that  one  does 
not  see  this  difference  in  color  without  a  polarizer  is  that  the  eye  observes  the 
resultant  of  the  rays  vibrating  in  both  directions,  and  only  when  one  set  of 

1  See  General  Bibliography  at  end  of  chapter. 

320 


ART.  266]  EXAMINATION  BY  PLANE  POLARIZED  LIGHT 


321 


rays  is  cut  out  can  it  be  perceived.     In  certain  minerals  the  absorption  is  so 
complete  in  one  direction  that  the  phenomenon  is  visible  without  the  nicol. 

266.  Uniaxial  Crystals. — The  colors  apparent  in  viewing  a  section  of  a 
colored  uniaxial  mineral,  cut  at  right  angles  to  the  optic  axis,  are  those  due 
to  the  ordinary  ray.  Since  these  travel  with  the  same  ease  in  every  direction, 
they  have  the  same  absorption  coefficient,  consequently,  no  matter  how  the 
stage  of  the  microscope  is  rotated,  the  color  remains  the.  same.  If  the  section 
is  cut  at  an  angle  with  the  optic  axis,  a  difference  in  color  may  appear  on  rotat- 
ing the  stage,  and  this  difference  is  at  its  maximum  when  the  section  is  cut 
parallel  to  the  optic  axis.  If  the  latter  section  is  placed  on  a  rotating  appara- 
tus, and  it  is  turned  in  altitude  about  the  optic  axis,  no  change  in  color  appears. 
In  other  words,  the  form  of  the  absorption  surface  is  that  of  an  ellipsoid  of 
rotation  with  the  optic  axis  as  its  axis.  The  optic  axis  is  also  the  direction  of 
least  or  greatest  absorption,  consequently  we  have  oblate  or  prolate  absorp- 
tion ellipsoids,  depending  upon  whether  this  axis  is  that  of  minimum  or 


FIG.  420.  FIG.  421.  FIG.  422. 

FIGS.  420  to  422. — Indicatrix  and  absorption  surface  compared  in  tourmaline,  apatite,  and  melilite. 
Solid  line  =  indicatrix,  broken  line  =  absorption  surface.  FIG.  420,  Negative,  absorption  w>e;  Fig. 
421,  negative,  absorption,  w<e;  FIG.  422.  positive,  absorption  co  <«. 

maximum  absorption.  The  two  cases  are  ordinarily  written,  Absorption 
O>E  (or  Absorption,  co>e),  and  Absorption,  E>O  (or  Absorption  e>co), 
where  O  and  E  represent  the  directions  of  vibration  of  the  ordinary  and  extra- 
ordinary rays,  and  co  and  e  the  directions  of  their  respective  indices.  Thus 
tourmaline  is  negative  (indices  e<  co)  and  it  is  darkest  when  the  vibrations 
take  place  at  right  angles  to  c  (c  =  e).  The  absorption,  therefore,  is  greatest 
parallel  to  the  direction  of  vibration  of  the  ordinary  ray  and  we  have  Absorp- 
tion co>e  or  Absorption  O>E  (Fig.  420).  In  apatite,  likewise  negative 
(e<co),  the  mineral  is  darkest  when  the  c  axis  lies  parallel  to  the  vibration 
direction  of  the  lower  nicol,  therefore  we  have  Absorption  e  >  co  (Fig.  421).  In 
melilite,  which,  in  some  cases,  is  positive  (e>  co),  the  extraordinary  ray  is  most 
absorbed  (Fig.  422),  therefore  Absorption  €>co.  Care  must  be  taken,  in 
writing  descriptions,  not  to  confuse  the  values  of  absorption  and  of  refractive 
indices;  the  word  absorption  should  always  be  written  before  the  former. 

Ordinarily,  in  uniaxial  crystals,  the  absorption  is  greatest  in  the  direction 
of  the  greatest  refractive  index  (Figs.  420  and  422),  whether  the  crystal  be 
21 


322  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  267 

positive  or  negative,  a  rule  first  given  by  Babinet1  who  recognized,  however, 
that  there  were  many  exceptions  to  it. 

267.  Biaxial  Crystals. — If  a  section  of  a  colored  biaxial  crystal,  cut  at 
right  angles  to  an  optic  axis,  be  examined,  it  will  be  found  that  the  absorption 
is  the  same  in  every  direction.  This  was  to  have  been  expected,  since  in  such 
sections  the  ease  of  vibration  is  likewise  the  same  in  every  direction.  If, 
however,  any  other  section  be  examined,  it  will  be  found,  in  many  cases,  that 
there  is  a  difference  in  color  in  two  directions,  and  that  the  colors  in  different 
sections  likewise  differ  one  from  another.  If  a  surface  of  absorption  be  con- 
structed, it  may,  in  general,  be  represented  by  a  triaxial  ovaloid,  the  values 
of  whose  axes  (absorption  axes,  Laspeyres2)  bear  no  relation  to  the  values  of 
the  axes  of  the  indicatrix.  It  was  formerly  held,  and  to  a  certain  extent  is 
still  so  taught,  that  the  absorption  axes  coincide  with  the  vibration  axes. 
They  do,  certainly,  in  uniaxial  crystals;  in  biaxial  crystals  in  which  the  vibra- 
tion directions  coincide  with  the  crystallographic  axes,  namely  in  the  ortho- 
rhombic  system;  and  along  the  b  axis  of  the  monoclinic  system.  Laspeyres3 
made  determinations  which  seemed  to  prove  that  the  absorption  axes  are, 
like  the  vibration  axes,  always  at  right  angles  to  each  other.  In  monoclinic 
crystals,  one  axis  coincides  with  crystallographic  b,  the  other  two  may  or 
may  not  coincide  with  the  directions  of  vibration.  Thus,  in  piedmontite 
he  found  that  the  absorption  axes  made  an  angle  of  20°  with  the  axes  of  vibra- 
tion. In  triclinic  crystals  none  may  coincide.  Voigt,4  Becquerel5  and  Ram- 
say6 came  to  the  same  conclusion,  but  Ehlers7  who  made  examinations  of 
certain  uniaxial  and  monoclinic  crystals,  the  latter  being  salts  of  cobalt, 
found  that  in  those  examined  the  absorption  axes  coincided  with  the  vibration 
axes. 

As  the  pleochroism  of  uniaxial  crystals  is  divided  into  two  classes,  so  may 
also  that  of  biaxial  crystals  be  divided,  according  to  whether  the  maximum 
absorption  lies  in  the  plane  of  the  optic  axes  or  at  right  angles  to  it. 

1  M.  Babinet:  Sur  Vabsorption  dans  les  milieux  colores  birefringents.     Comptes  Rendus, 
VII  (1838),  832-833- 

Idem:  Abstract  of  preceding.  Ueber  die  Absorption  in  farbigen  doppeltbrechenden 
Mitteln.  Pogg.  Ann.,  XLVI  (1839),  478-480. 

2  H.  Laspeyres:  Miner alogische  Bemerkungen.     Zeitschr.  f.  Kryst.  IV  (1879-80),  433- 
467,  in  particular  454. 

3  H.  Laspeyres:  Op.  cit.,  particularly  444-460,  and  especially  454-460. 

4  W.  Voigt:  Erklarung    der    Farbenerscheinungen     pleochroitischer    Krystalle.     Neues 
Jahrb.,  1885  (I),  119-141. 

6  Henri  Becquerel:  Sur  les  lois  de  Vabsorption  de  la  lumiere  dans  les  cristaux  et  sur  une 
methode  nouvelle  permettant  de  distinguer  dans  un  cristal  certaines  bandes  d' absorption  appart- 
enant  a  des  corps  dijferents.  Comptes  Rendus,  CIV  (1887),  165-169. 

6  W.  Ramsay:  Ueber  die  Absorption  des  Lichtes  im  Epidot  wm  Sulzbachthal.  Zeitschr.  f. 
Kryst.,  XIII  (1887-8),  97-134. 

7Iohannes  Ehlers:  Die  Absorption  des  Lichtes  in  einigen  pleochroitischen  Krystatten. 
Neues  Jahrb.,  B.  B.,  XI  (1897-8)  259-317. 


ART.  268]  EXAMINATION  BY  PLANE  POLARIZED  LIGHT  323 

268.  Pleochroic  Halos. — In  certain  minerals,  surrounding  small  inclu- 
sions of  other  minerals,  there  appear  rounded  spots  or  halos  which  are  more 
strongly  pleochroic  than  their  host,  although  the  maximum  absorption  direc- 
tions of  the  two  are  parallel.  As  a  matter  of  fact  the  "halos "  are  not  circular 
but  spherical,  for  they  show  the  same  outlines,  no  matter  what  the  direction 
in  which  the  section  cuts  the  mineral.  If  the  included  grain  is  decidedly 
elongated,  these  spots  are  ellipsoidal  instead  of  spherical,  though  such  occur- 
rences are  quite  rare.  The  borders  around  irregular  grains,  which  are  approxi- 
mately equidimensional,  are  also  spherical. 

The  minerals  in  which  these  pleochroic  halos  have  been  observed  are 
andalusite/augite,  biotite,1'  5  chlorite,1  cordierite,1  diopside,2  glaucophane,3 
hornblende,4'  5  ottrelite,1  and  tourmaline,6  and  the  inclusions  around  which 
they  occur  are  allanite,5  apatite,12  biotite,9  cassiterite,7  dumortierite,8 
pleonaste,9  rutile,6-  7  titanite,8  topaz,7  and  zircon.8 

As  to  the  cause  of  these  halos,  there  has,  until  recently,  been  great  diver- 
sity of  opinion.  They  have  been  thought  to  be  organic,  l>  4>  6  or  due  to  a 
local  increase  in  the  amount  of  the  iron  molecule,8'  10  but  within  the  last 
few  years  the  belief  has  become  general  that  they  are  due  to  radioactive 
emanations.11  That  they  are  not  due  to  diffusion  or  aggregation  is  clearly 
evident  from  the  fact  that  the  sphere  extends  across,  as  well  as  in  the 
direction  of  the  cleavage  in  such  minerals  as  biotite,  or  even  from  one 
mineral  to  another.  The  probability  that  the  halos  are  due  to  the  radio- 
active property  of  the  included  mineral  was  first  pointed  out  by  Joly,12 

1  H.  Rosenbusch:  Die  Steiger  Schiefer  und  ihre  Contactzone  an  den  Granititen  ion  Barr- 
Andlau  und  Hohwald.     Strassburg,  1877,  221,  281.* 

2  Idem:    Mikroskopische  Physiographic.     2te  Aufl.,  Stuttgart,  1885,  191. 

3  Konstantin    Anton  Ktenas:  Die  Einlagerungen  im  krystattinen  Gebirge  der  Kykladen 
anf  Syra  und  Sifnos.    T.  M.  P.  M.,  XXVI  (1907),  277. 

4  A.   Michel-Levy:    Propriete  optiques  des  aureoles  polychro'iques.     Comptes  Rendus, 
CIX  (1889),  973-976. 

5  E.  Cohen:    Ueber  pleochroitische  Hofe  im  Biotit.     Neues  Jahrb.,  1888  (II),  166-169. 
6H.    Traube:    Ueber   pleochroitische   Hofe   im    Turmalin.     Neues   Jahrb.,    1890    (I), 

186-188. 

7  H.  Rosenbusch:    Mikroskopische  Physiographic,  3te  Aufl.,  1892,  210. 

Johannsen  found,  in  a  topaz  granite,  a  pleochroic  band  at  the  contact  between  crystals 
of  biotite  and  topaz. 

8  A.   Michei-Levy:    Sur  les  noyaux  a  polychro'isme  intense  du  mica  noir.     Comptes 
Rendus,  XCIV  (1882),  1196-1198. 

Idem:  Proprietes  optiques  des  aureoles  polychro'iques.  Comptes  Rendus,  CIX  (1889), 
973-976. 

9  O.  Miigge:    Radio  aktivil  at  und  pleochroitische  Hofe.     Centralbl.  f.  Min.,  etc.,  1909,  66. 

10  Hj.  Gylling:    Nagra  ord  om  Rutil  och  Zirkon  med  sdrskild  hdnsyn  till  deras  sammanvax- 
tiing  med  Glimmer.     Geol.  Foren.  i  Stockholm  Forh.,  VI  (1882-3),  162-168. 

11  J.  Joly:  Radioactivity  and  Geology.     New  York,  1909,  64-69. 

12  Idem:  Pleochroic  halos.     Phil.  Mag.,  XIII  (1907),  381-383. 


324  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  269 

and  it  has  been  shown,  experimentally,  that  when  biotite1  or  cordier- 
ite-2'3  are  exposed  to  the  rays  of  a  small  particle  of  radium,  similar  colored 
and  pleochroic  patches  are  produced.  Another  evidence  for  this  theory  is  the 
fact  that  in  size  they  never  exceed  0.05  mm.,  and  average  0.04  mm.,  which 
is  almost  exactly  the  distance  that  radium  can  affect  a  photographic  plate 
through  a  medium  having  the  density  of  biotite.4  The  actual  change  pro- 
duced by  the  radium  in  the  mineral,  causing  this  intense  pleochroism,  is  not 
known.  Whatever  it  is,  it  causes  a  difference  in  the  double  refraction, 
which  may  be  greater  or  less  than  that  of  its  host,5  and  perhaps,  also,  a 
change  in  the  dispersion. 

269.  Pseudo-pleochroism,  Pseudo-dichroism,  or  Pseudo-absorption.— 

Certain  minerals  appear  colorless  in  one  direction  and  dirty  brown  in  another, 
giving  an  appearance  of  absorption,  although  the  phenomenon  is  not  due  to 
absorption  at  all.  According  to  v.  Fedorow,6  it  is  shown  to  some  extent  by 
all  minerals  which  have  strong  double  refraction,  especially  by  those  which 
have  very  good  cleavage  and  fine  lamellation,  as  calcite,  dolomite,  and  mag- 
nesite,  and  is  due  to  the  great  difference  in  the  refractive  indices  in  two  direc- 
tions, thus  permitting  the  rays  whose  vibrations  are  parallel  to  the  lamellation 
to  be  totally  reflected  and  those  which  enter  at  right  angles  to  it  to  pass 
through,  giving,  in  consequence,  an  appearance  of  partial  absorption.  Accord- 
ing to  Schroeder  van  der  Kolk,7  pseudo-pleochroism  is  due  to  the  fact  that 
innumerable  sub-microscopic  inclusions  are  arranged  in  parallel  position 
within  the  mineral,  so  that  when  light  enters  in  one  direction  it  passes  through 
without  change,  but  when  it  enters  in  another,  it  is  refracted  and  produces 
a  brown  tone. 

270.  Interference  Phenomena,  without  the  Analyzer,  Produced  by  an 
Overlying  Pleochroic  Mineral.8 — When  a  doubly  refracting  mineral  occurs 

1 J.  Joly :  PUochrok  holos.     Nature,  LXXVI  (1907),  589. 

2  O.  Mtigge:    Radioaktivitdt  als  Ursache  der  pleochroitischen  Hofe  des  Cordierit.     Cen- 
tralbl.  f.  Min.,  etc.,  1907,  397-399. 

3  Idem:    Radioaktivitat  und  pleochroitische  Hofe.     Centralbl.  f.  Min.,  etc.,  1909,  65-71, 
113-120,  142-148. 

4  Georg  Hovermann:     Ueber  pleochroitische  Hofe  in  Biotit,  Hornblende  und  Cordierit, 
und  ihre  Beziehungen  zu  den  a  Strahlen  radioaktiver  Elemente.     Neues  Jahrb.,  B.B.,  XXXIV 
(1912),  321-400. 

See  also  R.  J.  Strutt:  A  study  of  the  radio-activity  of  certain  minerals  and  mineral  waters. 
Nature,  LXIX  (1904),  473-475. 

Idem:  Same  title  as  preceding  Proc.  Roy.  Soc.,  London,  LXXIII  (1904),  191-197. 

6  E.  A.  Wiilfing:  Rosenbusch-W  ulfing:  Mikroskopische  Physiographic,  4te  Aufl.,  1904, 
347- 

6  E.  v.  Fedorow:    Pseudoabsorption.     Zeitschr.  f.  Kryst.,  XXXII  (1900),  128-130. 
Idem:  Ueber  Pseudochro'ismus  und  Pseudodichro'ismus.     T.   M.   P.  M.,  XIV  (1895), 

569-571- 

7  J.  L.  C.  Schroeder  van  der  Kolk:    SammL  Geol.  Reichsmuseum, Leiden,  VI  (1900),  89.* 

8  J.  L.  C.  Schroeder  van  der  Kolk:    Eine  eigenthiimliche  Folge  des  Plcochroismns  in 
Gesteinsschli/en.     Zeitschr.  f.  wiss.  Mikrosk.,  VII  (1890),  30-32. 


ART.  271J  EXAMINATION  BY  PLANE  POLARIZED  LIGHT  325 

underlying  a  thin  layer  of  a  strongly  pleochroic  mineral,  the  latter  acts 
as  an  analyzer  by  absorbing  the  rays  vibrating  in  one  direction,  and, 
as  a  consequence,  interference  colors  appear  in  the  thin  section.  If  the 
analyzer  is  inserted,  the  combined  minerals,  on  account  of  the  strong 
absorption  in  two  directions,  will  show  extinction  but  twice,  instead  of  four 
times,  on  a  rotation  through  360°. 

271.  Determination  of  Pleochroism. — Pleochroism  can  be  seen,  with 
the  unaided  eye,  in  but  very  few  minerals.  It  can  easily  be  seen,  under  the 
microscope,  by  permitting  only  rays  vibrating  in  one  direction  to  pass  through, 
as  by  inserting  the  polarizer  alone.1  In  this  way,  first  one  color  and  then 
the  other  can  be  observed  by  rotating  the  stage.  The  objection  to  this 
method,  which  is  the  one  usually  followed,  is  that  when  the  pleochroism  is 
very  slight,  the  eye  is  unable  to  perceive  it,  especially  when  the  stage,  and 


FIG.  423. — Dichroscope.     3/4  natural  size.     (Fuess.)  FIG.    424. 

not  the  polarizer,  is  rotated.     A  much  more  delicate  way  of  determining 
pleochroism  is  by  means  of  a  dichroscope  ocular. 

The  ordinary  dichroscope  (Fig.  423)  is  an  instrument  which  was  invented 
by  Haidinger.2  It  consists  essentially  of  a'calcite  prism  P  in  a  metal  casing, 
at  one  end  of  which  is  a  rectangular  opening  and  at  the  other  a  lens.  The 
length  of  the  calcite  is  so  chosen  that  the  two  images  of  the  rectangular 
opening  in  T  are  just  in  contact  with  each  other  (Fig.  424).  Since  one 
image  of  the  opening  is  produced  by  the  ordinary  ray,  and  the  other  by  the 
extraordinary,  the  vibration  directions  will  be  at  right  angles  to  each  other, 
consequently,  if  a  mineral  is  attached  by  a  bit  of  wax  over  the  opening  in  T, 
and  it  is  viewed  through  the  lens  L,  the  two  absorption  colors  produced  by 

1  Gustav  Tschermak:    Mikroskopische  Unterscheidung  der  Miner  alien  aus  der  Augit-, 
Amphibol-  und  Biotitgruppe.     Sitzb.  Akad.  Wiss.,  Wien,  LX  (1869),  5-16. 

2  W.  Haidinger:    Ueber den  Pleochroismus  der  Krystalle.     Pogg.  Ann.,  LXV  (1845),  1-30. 
See   also   V.    von  Lang:  Optische   Notizen:  Verbesserte   dichroscopische   Lupe.     Sitzb. 

Akad.  Wiss.  Wien,  LXXXII  (2),  1880,  174. 

Gustav  Halle:  Neues  vervollstdndigtes  Dichroskop.     Neues  Jahrb.,  1895  (II),  247-248. 

Idem:  Eine  neue  Form  des  Dichroskopes.     Zeitschr.  f.  Instrum.,  XV  (1895),  28. 

A.  Cathrein:  V  ervollkommung  des  Dichroskopes.     Ibidem,  XVI  (1896),  225-226. 

C.  Leiss:  Mittheilungen  aus  der  R.  Fuess' schen  Werkstdtte.  Verbindung  eines  Dichroskops 
mil  einem  Spectroskop.  Neues  Jahrb.,  1898  (II),  68-69. 


326  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  272 

the  vibrations  at  right  angles  to  each  other  will  be  seen  at  the  same  time.  By 
rotating  the  end  of  the  tube  T,  it  is  an  easy  matter  to  find  the  positions  of 
maximum  difference  in  absorption.  At  45°  from  this  position  the  two  colors 
will  be  the  same.  By  seeing  the  two  colors  thus,  side  by  side,  even  very 
slight  differences  in  absorption  can  be  observed. 

The  dichroscope  ocular1  (Fig.  425)  is  an  attempt  to  com- 
bine the  advantages  of  the  dichroscope  with  the  magnifying 
power  of  the  microscope  for  the  determination  of  the  pleo- 
chroism  of  small  mineral  fragments  in  thin  sections.  It  con- 
sists of  a  Huygens  ocular  in  which  there  is  inserted  a  cal- 
cite  prism  K,  beneath  which  is  a  diaphragm  with  a  rectan- 
gular opening.  As  in  the  ordinary  dichroscope,  the  length 
of  the  calcite  is  so  chosen  that  the  two  images  produced 
by  double  refraction  appear  side  by  side.  To  use  the  in- 
strument, both  analyzer  and  polarizer  must  be  removed. 
Sometimes  the  partial  polarization  of  the  light  by  the 
FIG  425—  Oc-  mirror  affects  the  results.  It  is  then  advisable,  if  possible, 
uiar  dichroscope.  to  tilt  the  microscope  backward  and  use  the  light  directly 
reflected  from  the  sky  or  clouds. 


272.  Determination  of  Hie  Absorption  Coefficient.  —  It  is  possible  to 
determine  the  values  of  the  coefficients  of  absorption  in  different  directions 
in  a  crystal,  but  this  belongs  rather  to  the  province  of  mineralogy  than  to 
petrology,  and  it  will  not  be  discussed  here.2  It  may  simply  be  mentioned 
that  quantitative  values  of  the  intensity  of  the  two  transmitted  rays  are 
obtained  by  a  combined  spectroscope  and  photometer,  either  a  Glan3  spec- 
trophotometer  or  a  Konigsberger4  microphotometer  being  used. 

1  C.  Leiss:  Mittheilungen  aus  der  R.  Fuess'schen  Werkstdtte.      Ocular-Dichroscop  fiir 
Mikroskope.     Neues  Jahrb.,  1897  (II),  92. 

2  See  Johann  Ehlers:    Die  Absorption  des  Lichtes  in  einigen  pleochroitischen  Krystallen. 
Neues  Jahrb.  B.  B.,  XI  (1897-98),  259-317. 

3  P.  Glan:     Ueber  ein  neues  Photometer.     Wiedem.  Ann.,  I  (1877),  351-360. 

Louis  Duparc  et  Francis  Pearce:  Traite  de  technique  mineralogique  et  petrographique,  I, 
Leipzig,  1907,  423-425. 

A.  E.  H.  Tutton:  Crystallography  and  practical  crystal  measurement.     London,  1911 
823-824. 

4  J.  Koenigsberger:  Ueber  ein  Mikro  photometer  zur  Messung  der  Absorption  des  Lichtes. 
Zeitschr.  f.  Instrum.,  XXI  (1901),  129-133. 

Duparc  and  Pearce:  Op.  cit.,  425-427. 

A.  E.  H.  Tutton:  Op.  cit.,  825-826. 


ART.  272]  EXAMINATION  BY  PLANE  POLARIZED  LIGHT  327 

GENERAL  BIBLIOGRAPHY 

Wilhelm  Haidinger:  Ueber  den  Pleochroismus  der  Krystalle.     Pogg.  Ann.,  LXV  (1845),  I-3O' 
Idem:  Pleochroismus  an  mehreren  einaxigen  Krystallen,  in  neuerer  Zeit  beobachtet.     Sitzb. 

Akad.  Wiss.  Wien,  XIII  (1854),  3-17. 
Idem:  Pleochroismus  an  einigen  zweiaxigen  Krystallen  in  neuerer  Zeit  beobachtet.     Ibidem, 

306-331. 
H.  de  Senarmont:  Versuche  iiber  die  kunstliche  Erzeugung  von  Polychro'ismus  in  krystallisirten 

Substanzen.     Pogg.  Ann.,  XCI  (1854),  491. 
Gustav  Tschermak:  Mikroskopische  Unterscheidung  der  Miner  alien  aus  der  Augit-,  Amphi- 

bol-  und  Biotitgruppe.     Sitzb.  Akad.  Wiss.  Wien,  LIX  (1869),  5-16. 
Viktor  v.  Lang:  Optische  Notizen.     Verbesserte  dichroskopische  Lupe.     Ibidem,  LXXXII 

(1880),  174- 

H.  Laspeyres:  Miner alogische  Bemerkungen.     Zeitschr.  f.  Kryst.,  IV  (1880),  444. 
Carl  Pulfrich:  Photometrische  Untersuchungen  uber  Absorption  des  Lichtes  in  anisotropen 

Medien.     Ibidem,  VI  (1882),  142-159. 
W.  Ramsay:  Ueber  die  Absorption  des  Lichtes  im  Epidot  wm  Sulzbachtahl.     Ibidem,  XIII 

(1888),  97-134- 
W.  Voigt:  Erklarung  der  Farbenerscheinungen  pleochroitischer  Krystalle.     Neues  Jahrb., 

1885  (I),  110-141. 

Er.  Mallard:  Sur  le  polychro'isme  des  cristaux.     Bull.  Soc.  Min.  France  VI  (1883),  45-52. 
Henri  Becquerel:  Sur  Vabsorption  de  la  lumiere  au  travers  des  cristaux.     Ibidem,  X  (1887), 

120-124. 


CHAPTER  XXII 
INTERFERENCE  COLORS 

273.  Interference. — As  we  have  already  seen,1  when  two  light  waves 
of  the  same  wave  lengths  and  in  the  same  plane  differ  by  half  a  wave  length, 
the  resultant  is  zero  and  the  light  is  extinguished.-    But  white  light  is  com- 
posed of  rays  of  many  different  wave  lengths  (Figs.  440-441),  and  the  condi- 
tions which  would  cause  a  difference  of  half  a  wave  length  for  one  color  would 
not  cause  it  for  another.     The  consequence  will  be,  naturally,  that  under  such 
conditions  the  light  seen  is  the  complementary  color  of  that  extinguished. 
We  have  also  seen  that  if  two  light  waves  differ  by  any  other  amount  than  half 
a  wave  length,  or  a  multiple  thereof,  the  resultant  wave  is  of  a  different  am- 
plitude from  the  original  wave.     Whether  the  resultant  of  the  combination 
of  several  waves  is  an  increase  or  a  decrease  in  the  amount  of  light,  the 
waves  are  said  to  interfere,  and  the  phenomenon  observed  is  spoken  of  as 
interference. 

274.  Color  of  Thin  Plates. — If  two  plates  of  glass  which  are  not  per- 
fectly true  planes,  such  as  panes  of  ordinary  window  glass,  are  pressed  to- 
gether, it  will  be  found  that  there  occur  certain  dark  spots  surrounded  by 
concentric  curves,  rather  far  apart  at  the    center   but    closer  and  closer 
together  toward   the   outer   rings.     The  colors,  from  the  center  outward, 
gradually  diminish  in  brightness,  and  the  outer  rings    approach   what  is 
known  as  "white  of   the  higher  orders."    By  pressing    the    glass    plates 
closer    together  the  inner   rings    broaden    and   the   whole  colored  series 
becomes  larger.     The  same  phenomenon  may  be  observed  if  a  piece  of 
thin  glass,  such  as  an  object-slip  or  a  cover-glass,  be  pressed  against  a 
glass  sphere  of  large  radius,  such  as  a  bell-jar  or  a  reading  glass.     In  this 
case,   owing  to   the  regularity  in  the    increase    in   thickness   of    the    air 
film  between  the  two  glasses,  the  curves  are  perfect  circles.     The  colored 
rings  observed  in  this  experiment  are  known  as  Newton's  rings,2  and  the 
series  of   colors,   from  the  center  outward,  as  Newton's  scale  of  colors. 
Less  symmetrically  distributed,  on  account  of  the  irregular  variation  in  the 
thickness  of  the  film,  are  the  colors  observed  in  a  soap  bubble,  or  in  a  film 

1  Art.  28,  supra. 

2  Sir  Isaac  Newton:  Opticks.     Reprinted  in  Klassiker  der  exakten  Wissenschaften, 
Nos.  96-97,  edited  by  W.  Ostwald.    Leipzig,  Book.  II,  Pt.  I. 

328 


ART.  274] 


INTERFERENCE  COLORS 


329 


M        Mi 


FIG.  426. — Passage  of  light  through  a  thin  film 
of  air  between  two  plates  of  glass. 


of  oil  spread  upon  the  surface  of  water.     The  colors  vary,  as  we  shall  see, 
with  the  thickness  of  the  film,  and  thus  is  produced  the  gradual  change  in 

color  of  a  soap  bubble,   which  reaches 
a  neutral  tint  just  before  breaking. 

Let  A  BCD  (Fig.  426)  represent  a 
thin  film  of  some  substance  (e.g.,  air) 
lying  between  two  films  of  a  medium 
having  a  higher  index  of  refraction 
(e.g.,  glass).  Any  ray  of  light,  such  as 
O,  upon  reaching  the  surface  of  differ- 
ent density  at  a,  will  be  partially  re- 
flected and  partially  refracted,  and  if 
a  represents  the  angle  of  incidence  OaN, 

|8  the  angle  of  reflection  Naa',  and  /z  the  angle  of  refraction  Maa",  we  will 
have,  since  the  light  is  passing  from  a  denser  to  a  rarer  medium, 
sin  a      i  .          sin  p. 

sin  /z     n ' '  n 

We  also  have,  since  the  angle  of  incidence  is  equal  to  the  angle  of  reflection, 

«  =  0.  (2) 

Consider  first  the  refracted  portion  of  any  ray  O.  Upon  reaching  the 
second  surface  of  the  film  BD,  it  will  again  be  partially  reflected  and  partially 
refracted  toward  b  and  a"'.  Since  the  two  surfaces  AC  and  BD  are  parallel, 
Maa"  =  aa"Ni  =  n,  and  since  the  angle  of  incidence  is  equal  to  the  angle  of 
reflection,  aa"Ni  =  Nia"b  =  /z.  Upon  reaching,  from  below,  the  surface  AC 
at  &,  the  reflected  portion  of  the  ray  a"b  will  again  be  partially  reflected  and 
partially  refracted  to  bf  and  b" .  Here  the  angle  a"bM 2  =  M2bb"  =  iz.  At  b 
the  portion  of  the  ray  a"b  refracted  upward  into  the  air  will  make  an  angle 
such  that 

sin  /z 

sine  angle  of  refraction  at  b 

for  the  light  is  now  passing  from  a  rarer  to  a  denser  medium. 
Transposing,  we  have 

sine  angle  of  refraction  at  b  = 

Uniting  equations  (i)  and  (3),  we  have, 

sine  of  refraction  at  b  =  sin  a, 
therefore  the  angle  of  refraction  at  b  (Nzbbf)  =  a,  or 

Consider  now  a  ray  O\,  parallel  to  the  ray  O.  When  it  reaches  the  film 
AC  at  b  it  will  likewise  be  partially  reflected  and  partially  refracted.  It 
makes  an  angle  of  a  with  the  normal  N2,  and  its  reflected  ray  b  makes  an 
angle  N->bb'  =  ft  =  a.  But  from  equation  (4)  the  angle  of  refraction  of  the 


sin  AZ 

n 


(3) 


330  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  275 

first  ray  at  b  is  also  «=/?,  therefore  the  two  rays  coincide  and  travel  along  the 
same  path  from  b  to  b'.  The  two  rays,  however,  have  traveled  different  dis- 
tances. When  the  ray  O  is  at  a,  the  ray  O\  is  at  x,  since  the  two  rays  are 
parallel  and  the  ray  front  ax  is  at  right  angles  to  them.  The  ray  Oi  continues 
on  at  the  same  rate  to  b,  but  the  ray  O,  passing  to  a  rarer  medium,  travels 
a  greater  distance  ae,  the  amount  depending  upon  the  density  of  the  medium. 
The  ray  O,  therefore,  which  must  travel  from  e  to  a"  to  b,  before  it  can  start 
on  the  path  traveled  by  Oi,  is  just  the  distance  ea"+a"b  behind  the  other, 
or,  as  we  say,  is  that  much  retarded.  The  consequence  is  that  the  two  rays 
traveling  along  the  same  path  W  are  in  different  phase,  the  lag  depending 
upon  the  thickness  of  the  film,  the  angle  of  incidence  cf  the  light,  and  the 
refractive  index  of  the  substances. 

Suppose,  for  the  moment,  that  monochromatic  light  having  a  wave 
length  of  \  is  used.  If  the  film  is  of  such  a  thickness  that  ea"-\-a"b  will  cause 

a  difference  of  phase  just  equal  to  -,  the  rays  will  so  interfere  that  complete 
darkness  is  produced. 1  If  the  thickness  of  the  film  is  somewhat  greater,  so 
that  the  retardation  is  — ,  — -,  — ,  etc.,  the  effect  is,  of  course,  the  same. 

22        2  7 

If  a  wedge-shaped  film  is  used,  instead  of  one  which  is  plane  parallel, 
there  will  be  successive  dark  bands  where  the  phase  difference  is  a  multiple 

of  -•     This  is  well  seen  in  the  experiment  of  Newton's  rings  of  which  mention 

was  previously  made.  Here  the  film  is  of  no  thickness  at  the  center,  where  the 
contact  is  good,  and  darkness  occurs.  Surrounding  this  there  is  a  succession 
of  bands  of  light  and  darkness,  the  latter  occurring  wherever  the  phasal 

difference  is  a  multiple  of  -• 

When  white  light  is  employed,  colors  or  colored  bands  will  be  seen  instead 
of  darkness.  This  is  due  to  the  fact  thai:  white  light  is  made  up  of  rays  of 
different  wave  lengths  (Figs.  440-441)  which  interfere  at  different  distances, 
and,  as  a  result,  the  color  seen  at  any  point  is  that  due  to  the  subtraction  of  one 
color  from  the  original  white  light.  This  is  beautifully  seen  in  soap  bubbles, 
in  which,  as  the  film  becomes  thinner  and  thinner,  successive  wave  lengths 
interfere. 

275.  Newton's  Color  Scale. — As  we  have  seen,  white  light  is  made  up  of 
many  rays  of  different  wave  lengths,  which  travel  with  different  velocities 
and  are  differently  refracted.  As  laid  down  by  Newton,2  the  colors  are 
divided  into  the  following  orders: 

1.  Black,  blue,  white,  yellow,  red. 

2.  Violet,  blue,  green,  yellow,  red. 

3.  Purple,  blue,  green,  yellow,  red. 

1  Art.  28,  supra. 

2  Sir  Isaac  Newton:  Opticks,  Bk.  II,  Pt.  I,  obs.  4. 


ART.  276] 


INTERFERENCE  COLORS 


331 


4.  Green,  dirty  red. 

5.  Greenish  blue,  red. 

6.  Greenish  blue,  pale  red. 

7.  Greenish  blue,  reddish  white. 

276.  Color  Scale  according  to  Quincke. — Since  Newton's  time,  the  color 
scale  has  been  worked  out  in  great  detail,  and  the  numerical  values  for  the 
retardations,  and  the  thicknesses  of  the  air  films  necessary  to  produce  the 
colors,  have  been  determined.1  The  values  obtained  by  different  observers 
are  not  all  alike,  owing  to  the  fact  that  the  positions  of  the  different  bands  in 
the  scale  vary  somewhat  for  different  modes  of  illumination.  Thus  the  values 
for  the  retardation  of  the  sensitive  violet  is  given  by  Wertheim  and  by  Quincke 
as  575MM,  by  Rollet  as  556^,  and  by  Kraft  as  535.6  to  557.6/z/x  for  clear  sky. 
In  most  of  the  petrographic  test-books,  Quincke's  values  have  been  given.2 
They  are  as  follows: 

NEWTON'S  COLOR  SCALE 
(Modified  from  Quincke) 


XT               Retardation 
A°-     !            x  =  S89 

1 

Order 

Interference  colors 
between 
crossed  nicols 

Interference  colors 
between 
parallel  nicols 

i 

2 

3 
4 

6 

8 
Q 
10 
ii 

12 
13 
14 
15 

16 

<W 
40 

97 
158 
218 
234 
259 
267 

275 
281 
306 
332 
430 
505 
536 
55i 

0 

Black  . 

Bright  white 
White  ^ 
Yellowish  white 
Brownish  white 
Brownish  yellow 
Brown 
Light  red 
Carmine 
Dark  reddish  brown 
Deep  violet 
Indigo 
Blue^ 
Grayish  blue 
Bluish  green 
Pale  green 
Yellowish  green 

Iron-gray  .  .  . 

"1/4" 

Lavender-gray  

Grayish  blue  
Clearer  gray  
Greenish  white  
Almost  pure  white  
Yellowish  white  
Pale  straw-yellow  
Straw-yellow  
Light  yellow  

"1/2" 

Bright  yellow 

Brownish  yellow  
Reddish  orange  
Red  

3/4 

Deep  red  

1  G.  Wertheim:  Memoir e  sur  la  double  refraction  temper  airement  produite  dans  les  corps 
isolropes,  et  sur  la  relation  entre  i'elasticite  mecanique  et  entre  V elastic-tie  optique.     Ann. 
Chim.  et  Phys.,  XL  (1854),  156-221,  in  particular  180. 

G.  Quincke:  Experimental-Unlersuchungen.  Ueber  Newton' sche  Farbenringe  u.  s.  w. 
Pogg.  Ann.,  CXXIX  (1866),  177-218,  in  particular  180. 

Alexander  Rollet:  Ueber  die  Farben  welche  in  der  Newton' schen  Ringsyslemen  aufein- 
anderfolgen.  Sitzb.  Akad.  Wiss.  Wien,  LXXVII  (1878),  3  Abth.,  177-261. 

Camille  Kraft:  Badania  doswiadczalne  nod  skalq  barw  inter jerencyjnych  (Etudes  experi- 
mentales  sur  I'echelle  des  couleurs  d'inlerrerence).  Bull,  internat.  de  1' Academic  des  Sci.  de 
Cracovie.  Cl.  sci.  math,  et  nat.  (Anzeiger  d.  Akad.  d.  Wiss.  Krakau.  Math-naturw.  Cl.) 
Cracovie,  1902,  310-353. 

2  A.   Michel-Levy:  Miner aux  des  roches,  64.     Groth:  Physikalische  Krystallographie, 
4te  Aufl.,  47.     Rosenbusch-Wiilfing:  Mikroskopische  Physiographic,  4te  Aufl.,  228.  Duparc 
et  Pearce:  Traite  d;  tech.  Min.   et   petr.,   182.     Johannsen:  Rock-forming  Minerals,   10. 
Winchell:  Elements,  etc.,  62-63. 


332 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  277 


NEWTON'S  COLOR  SCALE— (Continued) 
(Modified  from  Quincke) 


No. 

Retardation 

A  =  589 

Order 

Interference  colors 
between 
crossed  nicols 

Interference  colors 
between 
parallel  nicols 

1  7 

-ft- 

Purple  

Lighter  green 

18 
10 

575 

Violet  
Indigo 

Greenish  yellow 
Golden  yellow 

20 

21 

664 
728 

Sky-blue  
Greenish  blue  

Orange 
Brownish  orange 

2  2 

747 

Green  .  . 

Light  carmine 

27 

826 

Lighter  green  

Purplish  red 

24 

843 

Yellowish  green  

Violet-purple 

866 

Greenish  yellow  

Violet 

26 
27 

910 

048 

3/2 

Pure  yellow  
Orange  .  . 

Indigo 
Dark  blue 

28 

008 

Bright  orange-red  

Greenish  blue 

20 

IIOI 

Dark  violet-red  

Green 

•?O 

1128 

>-  Light  bluish  violet 

Yellowish  green 

7  T 

2 

Indigo   .  . 

Impure  yellow 

M  ro  rj-  t/ 
•J  CO  PO  CO  f 

1258 

1334 
1376 
1426 

Greenish  blue  
Sea-green  
Brilliant  green  
Greenish  yellow  

Flesh  colored 
Brownish  red 
Violet 
Grayish  blue 

36 
•37 

1495 

Flesh-color  
Carmine 

Sea-green 
Green 

38 
-2Q 

1621 



Dull  purple  
Violet-gray  

Dull  sea-green 
Yellowish  green 

40 

42 

1682 

I7H 

1744 

2 

Grayish  blue  
Dull  sea-green  
Bluish  green 

Greenish  yellow 
Yellowish  gray 
Lilac 

43 

i8n 

Light  green  

Carmine 

44 

1027 

Light  greenish  gray 

Grayish  red 

2OO7 

Whitish  gray  .... 

Bluish  gray 

46 

2048 

Flesh-red  

Green 

277.  Color  Scale  according  to  Kraft. — The  determinations  by  Kraft  were  made 
in  great  detail  for  illumination  by  Argand  lamp  or  incandescent  electric  light,  Auer 
burner,  electric  arc,  sunlight  reflected  from  snow,  gray  sky,  and  clear  sky.  He 
found  that  not  only  does  the  same  retardation  produce  different  colors  with  different 
methods  of  illumination,  but  the  widths  of  the  bands  of  color,  as  plotted  for  differ- 
ent retardations,  differ.  Thus  the  sensitive  violet  was  found  to  have  the  following 
retardations: 

Argand  or  incandescent  electric 576 .4  to  590.0 

Auer  burner 567 .  o       582 . 6 

Electric  arc '. .' 554.1       571.3 

Snow  illuminated  by  the  sun' 551 .  2       571.0 

Gray,  cloudy  sky 541 . 8       563 .  o 

Clear  sky 535-6       557-6 

For  comparison  there  are  given,  in  the  following  tables,  Kraft's  values  for 
clear  and  for  cloudy  sky,  the  conditions  under  which  microscopic  illumination  is 
most  commonly  obtained.  The  numbers  which  have  been  added  after  the  colors 
correspond  to  the  same  numbers,  as  nearly  as  it  is  possible  to  determine,  in  the  table 
in  Article  276. 

1  C.  Kraft:  Op.  tit. 


ART.  277] 


INTERFERENCE  COLORS 


333 


TABLE  OF  INTERFERENCE  COLORS,  LIGHT  FROM  A  CLEAR  SKY 
(According  to  Kraft)     i  =  5  5  o  ft  ft 


Order 

Interference  color,  nicols  crossed 

Retar- 
dation 

Interference  color,  nicols  parallel 

Order 

I 

Black    (i),    passing     through    iron- 
gray  (2)  to 

000.00 

The  color  of  the  source  of  the  light, 
passing  through  white  (7)  to 

I 

Lavender-gray  (3) 

Grayish  blue  (4) 

Yellowish  white  (n) 

Brown  (13) 

White,  tinted  with  greenish  blue  (6) 

Reddish  orange  (14) 

*>A  1     R 

Red  (15) 

White,  with  traces  of  greenish  blue 

Dark  carmine  (16) 

D                  1     (     } 

Greenish  white  (7) 

White  with  tint  of  yellowish  green 

Deep  violet  (18) 

II 

Light  greenish  yellow 

Indigo  (19) 

T  .  .          11        /     \ 

Brown  (13) 

Blue  (20) 

Orange 

473    I 

Reddish  orange  (14) 

488  8 

Greenish  blue  (21) 

Light  red  (15) 

Carmine  (16) 

rsiuisn  green 

Purple  (17) 

Green  (22) 

II 

Violet  (i  8) 

.       .     . 

i  eliowisn  green  (.24; 

617    5 

P         '  h       11        (     ) 

g 

Lrrccmsn  yciiow  ^25) 

Blue  (20) 

Yellow  (26) 

Orange  (27) 

Greenish  blue  (21) 

7-5  e     e 

Reddish  orange  (28) 

7ce    -a 

Light  red  (29) 

761   4 

Carmine  (29) 

1      green 

Green  (22) 

811   7 

Purple 

836   8 

Violet  (30) 

III 

Yeiiowisn  green  (.24; 

846    2 

T     A'          f       \ 

r>         •  u       11        i     \ 

885   3 

890  o 

Yellow  (26) 

047    i 

Blue 

Orange  (27) 

947   8 

Reddish  orange  (28) 

Greenish  blue  (32) 

Light  red  (29) 

Carmine  (29) 

1027   o 

rJiuisn  green  (33) 

Purple  (30) 

1088  o 

Green  (34) 

III 

Violet  (30) 

1  114    2 

T     A'           f        \ 

1  123   7 

i  euowisn  green 

. 

1166.1 

334 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  277 


TABLE  OP  INTERFERENCE  COLORS,  LIGHT  FROM  A  CLEAR  SKY—  (Continued) 
(According  to   Kraft)      X 


Order 

Interference  color,  nicols  crossed     j      Cation 

Interference  color,  nicols  parallel 

Order 

1161.4 
1166.1 
1192.  i 
1203.  o 
1223.5 
1232.  i 

1279-3 

1291.4 
1311.6 

1366.5 

1400.4 
1405-1 

1433-8 

1444-5 
I45I.O 
1468.7 
1503-6 
1535-0 
1550.3 
1570.0 
1628.6 
1659-5 

1691  .  2 

Blue 

Very  pale,  impure  yellow 

Greenish  blue  (32) 

Flesh  color  (36) 

Bluish  green  (32) 

Very  light  red 

Light  carmine  (37) 

Light  purple  (38) 

Green  (34) 

Yellowish  green 

Pale  violet  (39) 

Pale  indigo 

Greenish  yellow  (35) 

Very  pale,  impure  yellow 

Pale  blue  (40) 

IV 

Flesh  color  (36) 

Greenish  blue  (41) 

Light  red 

T31      '    -U                       I        \ 

Light  carmine  (37) 

Green  (43) 

Light  purple  (38) 

Yellowish  green  (44) 

Pale  grayish  violet  (39) 

Impure  grayish  indigo 

Greenish  yellow 

TABLE  OF  INTERFERENCE  COLORS,  LIGHT  FROM  A  GRAY  CLOUDY  SKY 
(According  to  Kraft)       A  =  55o//// 

Order 

Interference  color,  nicols  crossed 

Retar- 
dation 

I 
Interference  colors,  nicols  parallel        Order 

Black    (i),    passing    through    iron- 
gray  (2)  to 

o.o 
50.0 

IIO.O 

160.0 

222.2 
226.4 
236.8 
245.0 

250.2 
252.8 

263.7 
268.6 

277-3 
282.  I 
326.7 
347-5 
366.0 
4O6.O 

435-7 
472.9 

488.5 

The  color  of  the  source  of  the  light, 
passing  through  white  (7)  to 

I 
II 

Lavender-gray  (3) 

Grayish  blue  (4) 

Yellowish  white  (n) 

Brown  (13) 

White,    tinted    with   greenish  blue 
(7) 

Reddish  orange  (14) 

White      with    traces    of    greenish 
blue  (9) 

Red  (15) 

Dark  carmine  (16) 

Deep  purple  (17) 

Greenish  white  (10) 

White     with   a   tint   of    yellowish 
green  (u) 

Deep  violet  (18) 

Light  greenish  yellow 

Indigo  (19) 

Pale  yellow  (12) 

Blue  (20) 

Brown  (13) 

Greenish  blue  (21) 

Orange 

Reddish  orange  (14) 

ART.  277 


INTERFERENCE  COLORS 


335 


TABLE  OF  INTERFERENCE  COLORS,  LIGHT  FROM  A  GRAY  CLOUDY  SKY—  (Continued) 

(According  to  Kraft)     X  =  550*1** 


Order       Interference  color,  nicols  crossed 

Retar- 
dation 

Interference  colors,  nicols  parallel        Order 

Pale  red  (15) 

500.0 

508.3 
515-2 

531-0 
S4i.  8 
554-9 
563.0 

598.3 
617  .O 

677.1 
720.0 
735-3 
758.0 

767.9 
777-5 

808.6 
817-8 
836.6 
846.0 

880.4 
883.7 

922.8 
936.3 

942.8 
992.4 
I02O.O 

I03I-8 
1036.7 

1077-0 
1096.0 
III4.6 
II26.O 

II58.0 

1186.1 

I2O2.8 
I2I0.6 

1231.9 
1285.5 
1299.7 

1363.4 
1373.9 

1400.7 
1405.7 

1430.3 
1444.4 

1468.4 
1488.4 
1539.8 

1581.5 

1610.0 
1662.6 
1690  .  9 
1695.9 

i 

Bluish  green 

- 

Carmine  (16) 

Purple  (17) 

Green  (22) 

Yellowish  green  (24) 

II 

Violet  (18) 

Greenish  yellow  (25) 

Indigo  (19) 

Blue  (20) 

Yellow  (26) 

Greenish  blue  (21) 

Orange  (27) 

Reddish  orange  (28) 

Bluish  green 

Light  red  (29) 

Carmine  (29) 

Purple  (30) 

Green  (22) 

Yellowish  green  (24) 

Violet  (30)                                                        III 

Greenish  yellow  (25) 

Indigo  (31) 

Blue 

Yellow  (26) 

Greenish  blue  (32) 

Orange  (27) 

Reddish  orange  (28) 

Light  red  (29) 

Bluish  green  (33) 

Carmine  (29) 

Purple  (30) 

Green  (43) 

Yellowish  green 

III 

Violet  (30) 

Greenish  yellow  (35) 

Indigo  (31) 

Blue 

Very  pale,  impure  yellow 

Greenish  blue  (32) 

Flesh  color  (36) 

Bluish  green  (32) 

Very  light  red 

Light  carmine  (37) 

Green  (34) 

Light  purple  (38) 

Yellowish  green 

Pale  violet  (39) 

Greenish  yellow  (35) 

Pale  indigo 

Very  pale  impure  yellow 

Pale  blue  (40)        •                                         IV 

Flesh  color  (36) 

Greenish-blue  (41) 

Light  red 

Bluish-green  (42) 

Light  carmine  (37) 

Light  purple  (38) 

Green  (43) 

Yellowish-green  (44) 

Pale  grayish  violet  (39) 

Greenish-yellow. 

Impure  grayish  indigo 

CHAPTER  XXIII 
EXAMINATION  BETWEEN  CROSSED  NICOLS 

278.  Isotropic  Substances. — The  light  which  emerges  from  a  nicol  prism 
vibrates  parallel  to  one  plane  only,  that  is,  it  is  plane  polarized.     If  another 
nicol  is  placed  with  its  vibration  directions  at  right  angles  to  that  of  the  first, 
and  in  the  path  of  the  rays  coming  through  it,  all  light  will  be  cut  off  and  the 
field  will  appear  dark.     If  a  thin  section  of  a  colorless,  isotropic  substance  be 
placed  on  the  stage  of  the  microscope,  there  will  be  no  change  in  the  appear- 
ance of  the  field,  since  all  such  substances  permit  the  rays  to  vibrate  with 
equal  ease  in  every  direction,  consequently  it  will  have  no  effect  upon  the 
vibrations  of  the  light. 

279.  Anisotropic  Substances. — If  one  of  the  nicol  prisms  is  slightly 
rotated  so  that  it  is  not  at  right  angles  to  the  other,  a  small  amount  of  light 

A  will  be  found  to  pass   through.    Let  AA'  (Fig.  427) 

represent  the  vibration  direction  of  the  analyzer,  PP' 
that  of  the  rotated  polarizer,  and  OP  the  amount  of  light 
passing  through  the  latter.  Obviously,  the  light  can- 
not pass  through  the  analyzer  so  long  as  it  vibrates  in 
the  direction  PP',  but  it  may  be  resolved  into  two  rays 
vibrating  at  right  angles  to  each  other,  as  Oy  and  Ox. 
FIG.  427.— Passage  of  light  Of  these,  the  component  Ox  is  totally  reflected  by  the 

through  two  nicol  prisms.     ,,  _,  .-  ..  -i     •       -,  i  ,1 

balsam  film  of  the  upper  nicol  and  is  lost,  but  the 
component  Oy  will  pass  through,  the  amount  being  represented  by  the 
distance  from  O  to  y. 


PROBLEMS 

1.  On  the  stage  of  the  microscope  place  a  thin  section  of  a  colorless  or  pink 
garnet.     Note  that  the  mineral  permits  the  light  to  pass  through.  '  Insert  the  ana- 
lyzer.    The  field  now  remains  dark  during  a  complete  rotation  of  the  stage.     Try  to 
obtain  an  interference  figure.     If  a  uniaxial  figure  is  obtained  remove  the  mineral 
and  see  if  the  blank  slide  will  not  also  give  the  same  figure.     This  is  caused  by  the 
polarizing  effect  of  the'lenses  (Art.  356). 

2.  Place  a  basal  section  of  a  uniaxial  crystal  (e.g.,  calcite  or  quartz)  on  the  stage. 
Is  the  section  isotropic?    Is  the  mineral  isotropic?     Determine  the  latter  by  ob- 
taining an  interference  figure  (Chapter  XXIX).     On  comparison  with  an  isotropic 
mineral  we  see  that  the  basal  section  of  a  uniaxial  crystal  is  an  isotropic  section  of 
an  anisotropic  mineral. 

336 


ART.  282]  EXAMINATION  BETWEEN  CROSSED  NICOLS  337 

If  the  polarizer  and  analyzer  are  set  at  right  angles,  no  light  will  pass 
through,  as  we  shall  see,  unless  there  is  placed  upon  the  stage  of  the  micro- 
scope a  mineral  which  is  anisotropic.  Since  such  minerals  transmit  vibra- 
tions in  two  planes  at  right  angles  to  each  other,  it  follows,  if  the  mineral  is 
so  placed  upon  the  stage  of  the  microscope  that  its  vibration  directions  do 
not  coincide  with  those  of  the  nicols,  that  the  effect  is  the  same  as  though  the 
polarizer  were  placed  at  an  angle,  and  a  certain  amount  of  light  will  be  trans- 
mitted, the  amount  depending  upon  the  angle  which  the  principal  sections 
of  the  nicols  make  with  those  of  the  mineral.  Between  crossed  nicols,  there- 
fore, the  transmission  of  light  is  a  means  of  separating  anisotropic  crystals 
from  those  that  are  isotropic. 

280.  Retardation  in  Anisotropic  Media.  —  When  light  passes  through  an 
anisotropic  medium,  the  two  rays  into  which  it  was  broken  up  do  not  emerge 
at  the  same  time,  but  one  lags  behind  the  other.     This  retardation  depends 
upon  the  wave  velocities  and  the  thickness  of  the  section.    The  wave  veloci- 
ties themselves  are  dependent  upon  their  respective  refractive  indices  (n\ 

and  HZ),  and  vary  as  —  and  —  •    If  V\  is  the  greater  velocity,  and  Vz  the  lesser, 

Hi  H2 

then  —  >  —  and  HI<HZ,  n\  and  n2  being,  respectively,  the  refractive  indices 
n\     HZ 

of  the  rays  having  velocities  of  Vi  and  F2. 

Let  /i  be  the  time  for  the  ray  AI,  and  /2  the  time  for  Az  to  pass  through  the 
crystal.  If  M  is  the  thickness, 

T         ft£lf-       *"" 

M  =  t1—1h  =  n1M,  (i) 

»i    y 

and  M=tz  —  ,tz_=nzM.  (2) 

Whereby  the  retardation  R  is  given  by  the  equation 

R  =  tz-h  =  M(nz-nl}1  (3) 

a  result  given  in  millimeters  if  M  is  so  given. 

281.  Phasal  Difference.  —  If  X  is  the  wave  length  of  the  light  considered, 
the  phasal  difference  P  betwreen  the  two  emerging  rays  may  be  expressed  by 

»i) 

Uj 


282.  Interference  of  Polarized  Light.  —  The  interference  of  polarized 
light  was  studied  by  Arago  and  Fresnel1  who  found  (a)  that  two  rays  of  light, 
polarized  in  planes  at  right  angles  to  each  other  and  afterward  brought  into 
the  same  plane,  will  interfere  if  they  originally  belonged  to  the  same  beam  of 
polarized  light,  but  (b)  two  rays  of  light,  from  different  beams,  polarized  at 

1  Arago  et  Fresnel:  Memoir  e  sur  V  action  que  les  rayons  de  lumiere  polarisee  exercent 
les  uns  sur  les  autres.     Ann.  Chim.  et  Phys.,  X  (1819),  288-305. 
A.  Fresnel:  Oeuvres  completes,  Paris,  1866,  I,  521. 
22 


338 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  282 


right  angles  to  each  other,  may  be  brought  into  the  same  plane  of  polarization 
without  interfering. 

Let  PP'  (Fig.  428)  be  the  vibration  direction  of  the  polarizer  of  a  micro- 
scope, and  let  OP  and  OP'  each  represent  one-half  the  amplitude  of  the  vibra- 
tions of  the  light  emerging  from  it.  Let  UUf  and  VV  be  the  vibration  direc- 
tions in  an  anisotropic  crystal  lying  in  the  path  of  the  polarized  ray.  The 
plane  polarized  light,  upon  entering  the  crystal,  is  broken  into  two  rays  whose 
rate  of  transmission  is  not  the  same,  one  of  them  being  transmitted  with 
greater  difficulty  than  the  other.  As  a  result,  when  the  .two  rays  emerge, 
they  will  differ  by  some  part  of  a  wave  length.  Let  us  first  consider  the  case 
of  a  retardation  of  a  single  wave  length  (X)  or  a  multiple  thereof. 

Case  I.  The  vibrations  differ  by  X  or  a 
multiple  of  X  (N\),  and  the  nicols  are  parallel. 
Since  the  retardation  is  just  one  wave  length, 
the  two  components' emerge  at  O  at  the  same 
time.  One  component  will  start  toward  v,  the 
other  toward  u,  and,  owing  to  their  different 
amplitudes,  will  reach  these  points  at  the  same 
time,  the  locations  being  determined  by  the  re- 
solution of  forces.  If,  now,  an  analyzer,  with 
its  vibration  directions  parallel  to  that  of  the 
polarizer,  be  inserted  above  the  mineral,  the  two  components  will  be  brought 
back  into  the  plane  in  which  they  were  previously  vibrating.  The  ampli- 
tudes are  represented  by  Ox  and  Ow.  The  rays,  differing  by  N\,  have  their 
vibrations  in  the  same  direction,  hence  they  have  the  same  sign  and,  by 
their  interference,  the  resultant  is  equal  to  their  sum  (Ox+ Ow). 


FIG. 


FIG.  429. 


FIG.  430. 


N\ 


Case  II.     The  vibrations  differ  by  — ,  and  the  nicols  are  parallel.     If  the 

two  component  rays  emerge  with  a  difference  in  phase  of  — -,  where  N  repre- 
sents any  whole  number,  the  first  component  to  pass  through  has  already 
traveled  to  v  and  back  at  O  (Fig.  429)  when  the  second  component  arrives 
there.  When  the  first  has  reached  /,  the  second  has  just  reached  u.  Upon 


ART.  283] 


EXAMINATION  BETWEEN  CROSSED  NICOLS 


339 


inserting  the  analyzer  parallel  to  the  polarizer,  the  two  components  are 
brought  back  to  the  same  plane  and  are  represented  by  Ox'  and  Ow.  But 
here  the  two  components  are  in  opposite  directions  and,  in  consequence,  have 
different  signs.  The  resultant,  produced  by  their  interference,  is  —Ox'+Ow. 
Case  III.  The  vibrations  differ  by  N\  and  the  nicols  are  crossed.  As  in 
Case  I,  the  emerging  rays  reach  u  and  v  (Fig.  430)  at  the  same  time.  Upon 
inserting  the  analyzer  with  its  vibration  direction  (A  A'}  at  right  angles  to  that 
of  the  polarizer,  the  resulting  components  are  Ox  and  Ow,  and,  since  they  lie 
in  opposite  directions,  the  resultant  is  Ox— Ow  =  o.  That  is,  with  monochro- 
matic light  and  crossed  nicols,  the  interference  of  two  components  differing 
by  A^  will  produce  darkness. 

N\ 

Case  IV.     The  vibrations  differ  by  —  and  the  nicols  are  crossed.     As  in 

Case  II,  one  emerging  ray  reaches  u'  (Fig.  431)  when  the  other  reaches  v. 

Upon  inserting  the  analyzer  A  A',  the 

two    new    components   vibrate   in  the 

same    direction    and   the  resultant    is 

Ow+Ox. 

The  above  four  cases  prove  Fresnel's 
first  proposition. 

Case  V.  Observations  with  an  analy- 
zer only  inserted.  When  ordinary  light, 
which  vibrates  with  equal  ease  in  every 
direction,  passes  through  an  anisotropic 
crystal,  it  is  separated  into  two  rays 
vibrating  in  planes  at  right  angles  to  each  other.  If  an  analyzer  is  placed 
above  such  a  crystal,  the  rays  which  reach  it  from  below  are  made  up  of 
those  which  are  parallel  to  the  analyzer  and  those  which  are  at  right  angles 
to  it,  and  if,  at  one  instant,  the  interference  is  with  the  same  path  difference, 
the  next  it  is  with  the  opposite.  Since  the  changes  take  place  extremely 
rapidly,  we  can  perceive  no  interference  phenomena. 

This  case  proves  the  second  statement  of  Fresnel. 

283.  Extinction  Angles. — Place  a  thin  section  of  a  doubly  refracting 
mineral  on  the  stage  of  the  microscope  between  crossed  nicols  and  in  such  a 
position  that  its  vibration  directions  coincide  with  those  of  the  nicols  (Fig. 
432).  Let  Oa  represent  half  the  amplitude  of  the  light  entering  through  the 
polarizer  P'P.  Upon  reaching  the  anisotropic  crystal  the  tendency  is  for  the 
light  to  be  resolved  into  two  planes  at  right  angles  to  each  other,  Oa  and  Ob. 
Since  these  directions  are  parallel  to  those  of  the  nicols,  the  Ob  component  of 
OP  will  be  zero  while  all  of  the  Oa  component  will  pass  through  without 
change  of  direction.  Upon  reaching  the  analyzer,  whose  vibration  direction 
A  A'  is  at  right  angles  to  PP'}  the  tendency  is  again  to  resolve  the  light  into 


FIG.  431. 


340 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  283 


two  rays,  one  of  which  will  pass  through,  and  the  other  will  be  reflected  out. 
The  components  of  the  ray  Oa,  along  the  vibration  direction  of  the  analyzer, 
will  be  zero,  and  Oa  in  the  direction  at  right  angles  to  it.  But  the  OA  com- 
ponent is  the  only  one  which  is  permitted  to  pass  the  analyzer,  consequently 
the  field  of  the  microscope  will  be  dark  just  as  though  there  were  no  mineral 
on  the  stage,  or  as  though  the  mineral  were  isotropic. 

If  the  mineral  section  be  slightly  rotated  (Fig.  433)  so 
that  its  vibration  directions  form  angles  with  those  of  the 
two  nicols,  then  any  ray  of  light  from  the  lower  nicol,  hav- 
ing a  half  amplitude  OP,  will  pass  through  the  mineral  by 
means  of  vibrations  Ou  and  Ov,  or  Ou  and  Ov',  depending 
upon  the  phasal  difference  between  them.  Let  the  phasal 

N\ 
difference,  for  example,  be  —  (Figs.    433,  434,  435,  436), 

then  the  first  component  will  have  traveled  from  O  to  v  and 
back  to  0  when  the  second  component  emerges  at  O.  When  the  first 
is  at  i)'  the  second  is  at  u.  Upon  reaching  the  analyzer  A  A1 ',  each  of 
the  two  rays  is  again  resolved  into  two  others,  and  the  components 
passing  through  are  represented  by  Ow  and  Ox.  Moving  in  the  same  direc- 
tion, the  resultant  which  reaches  the  eye  is  represented  by  Ow-\-Ox.  The 
amplitude  of  this  resultant  governs  the  brightness  of  the  light  and,  as  may 
be  seen  by  Figs.  433  to  436,  it  is  at  its  maximum  when  the  inclination  of  the 
vibration  axes  of  the  mineral  to  those  of  the  nicols  is  45°.  Beyond  that  angle 


FIG.  433- 


FIG.  434- 


FIG.  435. 


FIG.  436. 


the  intensity  decreases,  since  the  major  part  of  the  light  falls  to  the  component 
Ov,  which  is  reflected  out.  If  the  thickness  of  the  crystal  is  greater,  so  that 
the  phasal  difference  of  the  two  original  components  is  N\,  (Cf.  Case  IV, 
supra),  then  the  vibrations  lie  in  opposite  directions  and  the  resultant  is 
zero  or  darkness. 

It  follows  from  this,  that  when  a  section  of  an  anisotropic  mineral  is 
placed  upon  the  stage  of  the  microscope  and  it  is  rotated  through  360°,  at 
the  four  positions  in  which  its  vibration  directions  coincide  with  those  of  the 
nicols,  there  will  be  complete  darkness.  At  all  intermediate  points,  the  min- 
eral will  permit  light  to  pass,  the  amount  depending  upon  the  angular  posi- 


ART.  284]  EXAMINATION  BETWEEN  CROSSED  NICOLS  341 

tion  of  its  vibration  axes  with  reference  to  those  of  the  nicols.     It  is  at  its 
maximum  when  the  angle  between  the  two  is  45°. 

We  have  seen  (Case  III,  Art.  282)  that  if  monochromatic  light  is  used  and 
the  difference  in  path  of  the  two  waves  is  iVX,  darkness  will  result.  If  we 
place,  then,  upon  the  stage  of  the  microscope  a  section  of  a  mineral  which  in- 
creases in  thickness  toward  one  end,  we  will  have  dark  bands  wherever  the 
phasal  difference  is  X  or  a  multiple  thereof,  and  between  these,  by  gradations, 
the  greatest  amount  of  light.  We  know  that  light  of  different  colors  is  of 
different  wave  lengths,  consequently  if  we  have  total  interference  for  one 
color  we  may  not  have  it  for  another.  If,  then,  in  the  above  experiments, 
white  light  be  used  instead  of  monochromatic,  and  the  thickness  of  the 
section  be  such  that  the  path  difference  is  just  a  wave  length  for  yellow  light, 
it  will  be  found  that  for  red  light,  whose  wave  length  is  greater,  the  difference 
in  path  is  less,  and  for  blue  light,  with  less  wave  length,  it  is  greater.  The 
consequence  is,  that  instead  of  black  bands  appearing  in  a  mineral  wedge 
wherever  the  wave  lengths  of  one  color  differ  by  N\,  we  will  have  a  series  of 
colored  bands,  and  at  any  point  that  color  will  predominate  which  is  nearest 

in  path  difference. 

284.  Passage  of  Monochromatic  Light  through  Two  Nicol  Prisms  and  a 
Mineral  Section. — Let  us  assume  that  a  beam  of  monochromatic  light  (O, 
Fig.  437)  strikes  the  lower  surface  of  a  nicol  prism.  Its  vibrations,  which 
took  place  in  every  direction,  will  now  be  reduced  to  a  single  plane,  that  is, 
it  will  be  polarized  in  the  vibration  plane  of  the  nicol.  The  amount  of  light 
which  passes  through  the  nicol  will  not,  however,  be  as  great  as  that  which 
entered  it,  since  the  component  vibrating  at  right  angles  to  the  vibration 
plane  of  the  polarizer  will  be  totally  reflected  by  the  film  of  Canada  balsam  in 
the  nicol,  and  will  be  lost.  Since  the  calcite  is  a  much  denser  substance  than 
the  air,  the  vibrations  will  be  reduced  both  in  amplitude  and  in  wave  length 

(*'). 

Upon  emerging  into  the  air  (cr),  the  amplitude  of  the  waves  increases, 
though  it  cannot  reach  that  which  it  possessed  before  it  entered  the  nicol, 
where  a  part  of  the  light  was  lost.  The  wave  length,  however,  again  becomes 
exactly  what  it  previously  was  in  air  (af)  and,  of  course,  the  color  remains  un- 
changed. When  light  travels  in  air  it  continues  without  change  in  the 
direction  in  which  it  started,  consequently  its  vibrations  are  here  parallel 
to  a  single  plane,  that  of  the  vibration  direction  of  the  polarizer. 

When  the  ray  of  light  enters  an  anisotropic  medium  it  is  broken  up  into 
two  rays  (d)  which  vibrate  in  planes  at  right  angles  to  each  other.  Since  the 
medium  is  denser  than  air,  the  amplitude  and  the  wave  length  are  both  re- 
duced (d'). 

When  these  two  components,  with  vibrations  in  planes  at  right  angles  to 


342 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  284 


each  other,  leave  the  crystal,  they  emerge  at  different  points,  and  if  we  had 
only  to  deal  with  a  single  ray,  each  component  would  pass  on  uninterruptedly. 

But  it  is  not  a  single  ray  of  light  which  we 
have  to  consider,  but  a  beam  made  up  of 
innumerable  rays  which  entered  the  lower 
nicol,  consequently  the  point  of  emergence 
of  either  of  the  two  components  of  the 
ray  O  will  also  be  the  point  of  emergence 
of  the  other  component  of  some  other 
ray,  as  O\.  We  have  here,  therefore,  a 
case  of  two  simple  harmonic  motions  at 
right  angles  to  each  other,  and  usually  of 
different  amplitudes  and  in  different 
phases,1  combined  with  a  uniform  motion 
of  translation  in  a  direction  at  right  angles 
to  the  plane  of  the  resulting  elliptical 
motion.  As  a  consequence,  the  two  com- 
ponents of  the  rays  O  and  Oi  will  unite 
to  produce  a  movement  of  the  particle 
along  a  helix  (<?')  of  elliptical  cross-section 
(e).  The  light  is  elliptically  polarized.2 

The  elliptically  polarized  beam,  upon 
reaching  the  analyzer,  is  again  plane  polar- 
ized, this  time  in  a  direction  at  right 
angles  to  its  former  position  (/).  Both 
amplitude  and  wave  length  are  reduced 
(/')  on  account  of  the  density  of  the  cal- 
cite.  The  amplitude  will  be  less  than  the 
amplitude  in  the  polarizer  (bf)  since  there 
is  now  not  so  much  light  entering,  but  the 
wave  length  (/"')  will  be  the  same.  Here 
again  the  component  of  the  entering  ray, 
which  is  at  right  angles  to  the  vibration  di- 
rection of  the  nicol,  is  reflected  out  and  lost. 
Upon  emerging  in  air  (gf)  the  ampli- 
tude of  the  wave  is  again  increased  but, 
since  part  of  the  light  was  lost  in  the 
analyzer,  it  cannot  have  the  amplitude  it 
previously  possessed.  The  wave  length  is 
the  same  as  it  was  in  air  throughout  (af, 


PIG.  437. — Passage  of  monochromatic 
light  through  two  nicol  prisms  and  a  min- 
eral section.  (The  wave  is  shown,  in  every 
case,  rotated  into  the  plane  of  the  paper.) 
The  vibration  at  a'  diagramatically  repre- 
sents the  intensity  of  all  of  the  compo- 
nents of  ordinary  light  vibrating  in  air, 
although  actually  vibrating  in  many 
planes,  as  at  a. 


e').     The  vibrations  lie  in  a  single  plane,  that  of  the  analyzer. 

1  Art.  27. 

2  Art.  77. 


ART.  285] 


EXAMINATION  BETWEEN  CROSSED  NICOLS 


343 


285.  The  Intensity  of  the  Emerging  Light  (After  Fresnel).1 — The  intensity  of 
the  emerging  light  may  be  determined  analytically  as  follows: 

Let  PP'  (Fig.  438)  be  the  vibration  direction  of  the  polarizer,  and  AA'  that  of 
the  analyzer.  To  make  the  problem  cover  all  cases,  let  the  angle  between  polarizer 
and  analyzer  be  any  angle  <?. 

Let  UU'  and  VV  be  the  vibration  directions  in  an  anisotropic  mineral  section, 
and  let  6  be  the  angle  which  UU'  makes  with  the  polarizer. 

Let  d  ( =  Od)  be  the  amount  of  light  coming  from  the  polarizer.     On  passing  into 
the  thin  section  it  is  broken 
up  into  the  two  rays  x  and 
y  (Ox  and  Oy  in  the  figure) . 

From  Fig.  438  we  have 

y 


cos  0  = 


d 


sin  e 


-j,  where- 
a 


=  d  sin  0,  and  y  =  d  cos  d. 
But  from  equation  (7),  Art. 

2ir 

25,   we  have  d  =  r  sin  •=-  t, 

in  which  d  is  the  displace- 
ment, r  the  amplitude,  / 
the  time  since  the  begin- 
ning of  the  movement,  and 
T  the  time  of  one  period. 
Substituting  this  value  fcr 
d  in  the  equation  above,  we  have 


FIG.  438. 


. 
sin  d  sin  - 


27T 

=  r  cos  0  sin  -~r/. 


(i) 


(2) 


Each  of  the  two  rays  now  moves  with  a  different  velocity,  and  independently, 
through  the  slide.  If  t\  and  tz  are  the  periods  of  time  required  for  the  two  rays  to 
pass  through,  the  value  of  the  displacement,  upon  emergence,  will  be  that  given  by 
equation  (8),  Art.  25,  and  equations  (i)  and  (2)  become 


x  =  r  sin  0  sin 

2 

y'  =  r  cos  0  sin  -    ^    2  • 

But  /i  and  /2  are  equal  to  n\M  and  w2Af  (Eq.  i  and  2,  Art.  280),  therefore 

x'  =  r  sin  0  •  sin' 

y  =r  cos  0  •  sin- 


(3) 
(4) 

(5) 
(6) 


These  are  the  equations  of  the  amplitude  of  the  rays  after  leaving  the  mineral  sec- 
tion. 


1  A.  Fresnel:  Oeuvres  completes,  I,  615. 


344  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  285 

The  rays  now  pass  through  air  and  reach  the  upper  nicol,  where  they  are  again 
resolved  into  a  single  plane  with  components  w  and  z,  or  w  and  z'  (Ow,  Oz,-0z'  in 

Nl 

the  figure),  depending  upon  whether  the  difference  of  phase  is  —  or  Nl. 

The  equations  are  in  the  same  form  as  before,  as  may  be  determined  from  the 
•figure. 

w  =  y'  cos  (8  —  <p)  =r  cos  6  -  cos  (0  —  <p]  •  sin  —  :  —  =,  --  >  (7) 


.    2ir(t  — 

z  =  x  sin  (0  —  <f>)  —  r  sin  e  -  sin  (6  —  <p)  -  sin  —  :  —  ~  —  (8) 

But  these  two  vibrations  are  now  polarized  in  the  same  plane,  consequently 
they  interfere.  The  amplitude  A  is  equal  to  the  amplitudes  of  the  two  components, 
therefore 


.  N      .  . 

A  =w=±z  =  r  cos  0  •  cos  (d  —  <p)  •  sin  —  :  —  ™  ---  =*=r  sin  e  •  sin  (6  —  <p)  •  sin  —  -  —  ~  — 

For  simplicity,  substitute  K  —  r  cos  9  •  cos  (8  —  <f>),  and  K'  =  r  •  sin  $  •  sin  (0  —  v>)  and 
the  equation  becomes 

„     .      27r(/-W2M)        „,     .      2*(t 

A  =  K  sin  -  —  ,  --  L±K'  sin  - 


By  comparing  equations  (8)  and  (9),  Art.  25,  with  equations  (8)  and  (7)  above, 
we  see  that  r±  and  rz  of  the  equation  for  the  amplitude  of  the  resultant  of  two  har- 
monic motions  (Eq.  4,  Art.  28)  become,  after  passing  through  the  two  nicols  and  the 
anisotropic  medium,  r  cos  6  cos  (0  —  <?),  and  rsin  6  sin  (O  —  v),  consequently  equation 
(4),  Art.  28,  may  be  written 

A2  =  [r  cos  e  •  cos  (8-<p)]2+[r  sin  6  -  sin  (0-v?)]2 

+  2J>  cos  6  -  cos  (0  —  <?)][>  sin  0-  sin  (0  —  <p)]cos  -  ^T~  (IZ) 

From  trigonometry  we  have 

\-ir(n2-nJM-]  ,fv(M2-»i)Jlf-| 

cos  2   -    —  ^  —        =1  —  2  sin2        —  ^  — 

Developing  equation  (n),  substituting  these  values,  and  inserting  /  for  A2, 
since  the  intensity  of  light  is  proportional  to  the  square  of  its  amplitude,  we  have: 

I  =  Az  =  r   cos2  (0  —  v)  -  cos2  0+r2  sin2  (0  —  <p)  -  sin2  0 
+  2r2  cos  (Q  —  <p)  -  sin  (0  —  <p)  •  cos  0  sin  0 


,  •  o  /       s     .          •  ,      «2—  wi\ 
—  r2sm2  (0  —  v?)  .  sin  20-  sm2N        _,      -j 

=  r2  [cos  (0  —  v)  •  cos  0  —  sin  (0  —  <?)  •  sin  0]2 

„  /7r(w2  —  Wi)M"\  I 
-[sin  2  (0-^)  .  sin  2  0  •  sin*/-  -i^,—  -1 

=  r2[cos2  ^-sin  2  (0-^,  -  sin  20  .  sin2/^2"^1^)  1-  (12) 


This  is  the  equation  for  the  intensity  of  light  after  passing  through  two  riicol  prisms 
and  a  mineral  section. 


ART.  285]  EXAMINATION  BETWEEN  CROSSED  N I  COLS  345 

.From  equation  (10),  Art.  25,  we  have 

vT  =  \. 
If  we  consider  the  velocity  of  propagation  (t?)  as  unity,  this  equation  becomes 

T  =  \.  (13) 

That  is,  the  time  of  oscillation  is  equal  to  the  wave  length.     Substituting  this  value 
in  equation  (12)  we  have 


Considering  this  equation  in  its  relation  to  the  positions  of  the  nicol  prisms,  we 
find  that  we  have  three  cases. 

Case  I.  The  vibration  directions  of  polarizer  and  analyzer  make  any  angle  (v) 
with  each  other.  In  equations  (13)  and  (14)  the  first  term  varies  with  the  angle 
between  the  vibration  directions  of  polarizer  and  analyzer,  and  the  second  depends 
upon  the  wave  length  of  the  light  used. 

If  tp  has  any  fixed  value,  and  the  thickness  of  the  section  M  remains  the  same, 
the  value  of  the  intensity  of  the  light  will  depend  upon  the  valire  of  0,  the  angle 
which  the  vibration  directions  of  the  section  make  with  that  of  the  polarizer.  When 

0  =  o,  <p,  -t,  or  <p-}—t,  the  second  member  becomes  zero,  and  the  intensity  of  the  light 

is  given  by 

7  =  r2cos2*,.  (15) 

Case  II.     The  vibration  directions  of  polarizer  and  analyzer  are  perpendicular  to 

each  other.    In  this  case  ^  =  oo°  =  ->     Substitute  this  value  in  (14)  and  we  have 

2 


/      T\ 
1  8—  -J  •  si 


.   9 
sin  20  'sm2 


;  — 
But,  by  trigonometry,  sin  2(0  —  ^)  =  —  sin  2  0,  whereby 

(16) 


Here  again  the  value  of  the  intensity  depends  upon  the  value  of  0,  the  inclination 
of  the  principal  sections  of  the  mineral  to  those  of  polarizer  and  analyzer. 

Equation  (16)  reaches  its  maximum  when  sin2  20=  i,  that  is  when  the  principal 
vibration  directions  of  the  thin  section  make  an  angle  of  45°  with  those  of  the  nicols. 
Its  minimum  value  is  reached  when  sin2  20  =  o,  that  is,  when  the  principal  vibration 
directions  of  mineral  and  nicols  coincide.  It  follows,  therefore,  that  the  field  must 
become  dark  four  times  on  rotating  the  stage  through  360°,  and  likewise  bright 
the  same  number  of  times  when  the  crystal  is  turned  45°  from  the  positions  of 
darkness. 

As  here  considered,  the  wave  length  X,  the  thickness  of  the  section  M,  and  the 
retardation  M(n%—  «i),  were  taken  as  of  fixed  values.  When  they  vary,  the  value 

of  sin2  (  —  -  J   varies,  consequently  the  intensity  of  the  light  also.     When 


346  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  286 


S1IV 


(•  2  —  )  =o  the  intensity  is  at  its  minimum,  since  then  equation  (16) 
becomes  zero  also,  and  the  field  is  dark.  This  result  is  produced  when  (n<i—n\)M  = 
2N-,  where  N  is  an  integer.  In  other  words,  the  field  is  dark  when  the  phasal 

2 

differ2nce  is  a  whole  number,  for  by  equation  (4)  Art.  281, 


a  result  like  that  derived  by  graphical  methods,  Case  III,  Art.  282. 

The  intensity  is  at  its  maximum  when  sin2  f  —  —  J  has  its  maximum  value. 

This  occurs  when  (nz—ni)M=(2N-}-i)-,  that  is,  when  one  of  the  light  waves  pre- 
cedes the  other  by  half  a  wave  length,  for 


which  is  the  same  result  as  that  derived  by  graphical  methods,  Case  IV,  Art.  282. 
Case  III.     The  vibration  directions  of  polarizer  and  analyzer  are  parallel.     In 
this  case  <f>  =  o,  whereby  cos2  ^  equals  unity,  and  equation  (14)  becomes 

/-r«i-sm»  20  •  sin' 


The  light  has  its  maximum  intensity  when  0  =  o,  -,  IT,  —  ,  or  271-,  and  its  minimum 

value  when  0  =  ->  —  »  —  »  or  —  •      That  is  to  say,  the  light  is  at  its  maximum  when  the 

444          4 
vibration  directions  are  parallel  or  at  right  angles  to  those  of  the  nicols,  and  at  its 

minimum  when  they  are  at  45°. 

The  relation  of  sin2  f  .      —  j   to  interference  may  be  demonstrated  as  in 

the  previous  case.     It  will  be  found  that  the  intensity  of  the  light  is  at  its  maximum 
when  —      .    Hl=N,  and  at  its  minimum  when  it  equals  —      —  ,  for  in  these  posi- 

A  2 

tions  7  =  r2(i—  o)  =  r2,  and  I  =  rz(i  —  i)=o,  results  which  are  the  same  as  those 
obtained  graphically  in  Cases  I  and  II,  Art.  282. 

286.  Two  Superposed  Mineral  Plates. 

/.  Vibration  directions  are  parallel. 

a.  Slow  rays  parallel, 

b.  Fast  rays  parallel. 

The  values  of  the  vibration  directions  in  any  mineral  section  are  expressed  by 
a  corresponding  section  through  the  ease-of-vibration-  or  the  Fresnel  ellipsoid. 
The  form  of  this  section  is  elliptical,  consequently  the  fastest  and  slowest  rays  are 
represented  by  its  principal  axes.  For  brevity,  the  terms  "fast-  and  slow-ray" 
will  be  used  hereafter  to  express  the  directions  of  greatest  and  least  ease  of  vibration 


ART.  286] 


EXAMINATION  BETWEEN  CROSSED  NICOLS 


347 


in  any  mineral  section.  These  terms  do  not  necessarily  mean  the  maximum  and 
minimum  values  in  a  mineral,  but  simply  the  maximum  and  minimum  in  the  particu- 
lar section  under  consideration. 

The  retardation  produced  by  an  anisotropic  mineral  is  given  by  equation  3, 
Art.  280, 


in  which  M  is  the  thickness  of  the  section,  and  HI  and  «i,  the  maximum  and  minimum 
refractive  indices  in  that  section.  If  another  thin  section  is  placed  above  the  first 
in  such  a  position  that  their  vibration  directions  are  parallel,  the  effect  will  be  that 
of  adding  or  subtracting  a  certain  amount  of  light  to  the  former,  depending  upon 
whether  the  vibrations  take  place  in  the  same  or  in  opposite  phase.  Thus  if  M." 
and  Mr  be  the  thicknesses,  and  n"2  and  «"i,  and  n'*  and  ri\  (Fig.  439)  the  refractive 
indices  of  the  two  minerals,  equation  (16)  will  become 

,  .  ,/TT  [Mr  (n'i-n'i)  =*  M"  (n\-n\}}\  .    . 

/  =  r2sm220-sm2^  -  ^  —  —)•  (23) 

If  the  fast  rays  of  the 
two  minerals  are  parallel, 
the  effect  will  be  axlditive, 
if  in  opposite  directions, 
subtractive,  the  two  acting 
as  a  single  mineral  plate,  as 
was  first  shown  by  Biot.1 
If  M"(ri'*-ri'd  =  M'(n',- 
n'i),  the  last  part  of  equa- 
tion (23)  becomes  sin2  0  =  0, 
consequently  the  value  of 
/  =  o,  or  darkness.  If  the 
retardations  of  the  two  are 
not  the  same,  the  result  is 
an  increase  or  a  decrease  in 
brightness,  the  amount  de- 
pending upon  the  two 
values.  If  the  retardation 
of  one  plate  is  known,  it 
serves  as  a  measure  of  that 
of  the  other,  and  also  to  determine  the  positions  of  its  fast  and  slow  rays. 

II.  Vibration  directions  of  minerals  are  at  other  angles  than  o°  or  90°  with  those  of 
the  nicols. 

Let  PP',  Fig.  439,  be  the  vibration  direction  of  the  polarizer,  and  A  A',  that  of 
the  analyzer,  and  let  the  angle  between  them  be  90°.  Let  n'z  and  n\t  and  n"z  and  n"\ 
be  the  fast  and  slow  rays  of  two  mineral  sections  making  angles  of  6'  and  0",  which 
are  neither  o°  nor  90°,  with  the  polarizer.  Let  Op  represent  the  amplitude  of  the 
ray  emerging  from  the  polarizer.  Upon  reaching  the  first  mineral  section  tt'2«'i,  it 
is  broken  up  into  two  waves  vibrating  parallel  to  On'  2  and  On'\,  and  with  amplitudes 
of  Ox\  and  Oyi.  Each  of  these  rays  is  broken  up  into  two  others  in  the  second  min- 

1  J.  B.  Biot:  Traile  de  physique.     Paris,  i8i6,TV,  419-422. 


PIG.  439. — Intensity    of   light   passing   through    two   superposed 
mineral  plates  between   crossed  nicols. 


348 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  287 


eral  section  w"2«"i;  the  ray  Oyi  into  Oyz  and  Ox'2,  and  Oxi  into  Oy'z  and  Ox«.  Upon 
the  insertion  of  the  analyzer,  each  of  these  four  rays  is  again  broken  up  into  two, 
but  of  these,  only  those  parallel  to  the  vibration  direction  of  the  analyzer  can  pass 
through,  those  at  right  angles  being  reflected  out,  consequently  the  light  reaching 
the  eye  is  represented  by  Om-\-Om' -\-Ow' — Ow. 

Analytically  the  same  result  may  be  obtained  in  a  manner  similar  to  that  used 
for  the  determination  of  the  intensity  of  the  light  emerging  from  two  nicols  and  a 
single  mineral  section.  The  demonstration  is  long  and  of  little  importance  in 
petrographic  work  except  when  the  angles  6'  and  6"  are  45°  and  135°,  in  which  case 
the  two  mineral  sections  are  at  right  angles  and  the  equation  takes  the  form  of  equa- 
tion (23).  If  the  nicols  are  parallel  and  the  minerals  at  an  angle,  the  equation, 
similar  to  (19)  is 

(  "\ff     ( ^.f     „'    \    ±     TUT    (>f     I 


If  the  minerals  are  rotated  until  their  vibration  directions  coincide  with  those  of  the 
nicols,  0  =  o°  or  90°,  sin220  =  o,  and  the  equation  becomes  7  =  r2. 


oon 


600 


400 


400 


287.  Examination  by 
White  Light.  Interference 
Colors. — As  shown  by  Case 
III,  Art.  282,  and  Case  II, 
Art.  285,  when  monochro- 
matic light  is  used  and  the 
thickness  of  the  mineral  sec- 
tion is  such  that  the  two 
waves  emerge  with  a  retarda- 
tion of  N\  the  stage  appears 
completely  dark  between 
crossed  nicols.  If,  now, 
there  be  used  white  light, 
which  is  composed  of  many 
rays  of  different  wave  lengths 
(Figs.  440-441),  the  wave 
length  of  a  certain  color  may 
be  such  as  to  produce  dark- 
ness, but  the  other  colors 
will  pass  through  with  greater 
or  less  intensity,  and,  as  a 
result,  they  will  produce  an 
interference  color.  This  may 
be  demonstrated  by  equation 

14,  Art.  285,  in  which  cos2  <p,  being  independent  of  the  wave  length  of  light 
or  of  the  thickness  of  the  section,  will  be  the  same  whether  monochromatic 
or  white  light  be  used.  In  the  general  case,  where  the  nicols  are  not  crossed, 


:too 


White      "3 


Black 


100. 


FIG.  440. 
FIGS.  440  and  441.- 


FIG.  441. 
-Wave  lengths  for  different  colors. 


ART.  287]  EXAMINATION  BETWEEN  CROSSED  NICOLS  349 

the  result  will  be  affected  by  sin  2  <p  -  sin  2  (6—  <p)  -  sin2  (-    2—^~    — J  of  which 

the  first  part  may  be  neglected,  since  these  values  affect  chiefly  the 
intensity  of  the  color.  They  produce  a  slight  dispersion  of  the  bisec- 
trices, but  this  is  so  small  that  the  result  will  be  but  little  affected 
by  their  omission.  The  second  part  of  the  term,  as  we  saw  in  Case 
III,  Art.  285,  became  o  when  P  =  N  (equation  17,  Art.  285)  for  any  defi- 
nite color,  that  is,  definite  wave  length  of  light.  As  a  result,  when  white 
light  is  used,  that  color  for  which  the  thickness  of  section  is  the  proper 
one  will  be  extinguished,  leaving  the  others  to  give  color  to  the  slide.  It 
was  shown  by  equation  18,  which  was  derived  from  an  equation  for 
crossed  nicols  but  which  might  just  as  well  have  been  deduced  from  equa- 
tion 14,  that  when  for  some  definite  color  P= ,  the  light  is  at  its  maxi- 
mum; consequently,  in  the  color  resulting  from  white  light,  that  color  whose 
phasal  difference  is  nearest  this  value  will  have  the  greatest  influence  upon  it, 
and  the  one  whose  phasal  difference  equals  AT  will  have  the  least,  for  it  will  be 
totally  extinguished.  Suppose  we  start  with  a  thin  section  which  is  of  such  a 
thickness  that  yellow  light  (D),  whose  wave  length  is  590^  (Art.  258),  is 
completely  extinguished.  Its  phasal  difference  must  then  be  N- 590^1*.  If 
white  light  be  used,  orange  and  green  are  a  little  more  and  less  retarded,  and 
red  and  blue  still  more  and  less.  Their  intensities  are  less,  since  they  differ 

by  some  amount  from  -      — X,  and  the  greater  this  difference  the  less  effect 

will  the  color  have  upon  the  resulting.  In  the  particular  case  taken,  with  a 
retardation  of  590^1^1,  the  resulting  color  is  indigo,  Jthe  complementary  color 
of  that  extinguished.  Again  if  R  =  4861*1*  (blue,  F),  the  resulting  comple- 
mentary color  is  orange.  If  the  phasal  differences  of  all  the  colors  ap- 
proach that  of  the  color  which,  in  the  particular  section  under  considera- 
tion, is  totally  extinguished,  then  more  and  more  are  the  colors  extin- 
guished until,  with  a  retardation  of  o/i/i,  darkness  is  produced.  This 
takes  place  in  isotropic  crystals  or  basal  section  of  uniaxial  crystals,  for 
in  such  the  velocities,  and  consequently  the  retardations,  are  equal  in  all 
directions. 

The  color  nearest  white  will  appear  when  the  least  light  is  extinguished. 
This  must  occur  when  the  phasal  difference  of  each  of  the  rays  is  nearest  the 

mean  of  all  values  of X,  relative  intensities  of  the  various  colors  being 

taken  into  consideration.  The  value  of  this  mean  is  about  250^,  conse- 
quently when  the  retardation  is  of  this  amount,  the  interference  color 
between  crossed  nicols  is  practically  white  (Fig.  440).  With  a  retardation  of 
575MM  the  most  intense  light,  near  the  D  line,  is  extinguished,  and  the  inter- 
ference color  remaining  is  a  violet,  known  as  the  sensitive  tint,  sensitive 
violet,  sometimes  "red"  of  the  first  order,  because  a  very  slight  change  in 


350 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  287 


retardation  in  either  direction  produces  a  marked  change  in  the  interference 
color,  red  on  one  side  and  blue  on  the  other. 

Since  the  values  of  maximum  and  minimum  intensity  are  exactly  reversed 
when  parallel  nicols  are  used,  the  interference  color  is  white  when  the  retarda- 
tion is  zero,  black  when  it  is  250^,  and  elsewhere  the  color  whose  wave  length 
equals  the  retardation.  The  colors  are  complementary  to  those  produced  by 
crossed  nicols. 

The  effect  of  the  superposition  of  several  mineral  plates  with  vibration 
directions  parallel  or  at  right  angles  may  be  determined  by  equation  23 
and  by  comparison  with  the  table  in  Art.  276.  Thus  if  the  retardation  of  one 

plate  is  575MM  and  the  interference  color 
is  violet,  and  of  another  plate  158/z/z  and 
the  color  gray,  the  resultant  of  the  two 
plates  in  parallel  position  is  a  retarda- 
tion of  575/*H-i58MM=733MM,  or  when 
the  vibrations  are  at  right  angles  575MM 
—  I58ju/i=4i7/iju.  In  the  first  case  the 
interference  color  is  greenish  blue,  in  the 
second  yellowish  brown. 

The  same  effect  as  with  minerals  in 
parallel  position  is  obtained  by  increas- 
ing the  thickness  of  a  section.  Thus  while  the  retardation  in  a  section  of 
quartz  0.03  mm.  thick  and  cut  parallel  to  crystallographic  c,  is  270^  and 
the  interference  color  is  a  pale  yellow,  in  a  section  0.06  mm.  thick  it  is  540^ 
and  the  interference  color  is  red.  Again  a  section  of  augite  0.02  mm.  thick 
has  a  retardation  of  500^  and  a  red  interference  color,  while  one  of  0.06  mm. 
has  a  retardation  of  isooju/z  and  is  of  a  third  order  yellowish  red. 

The  effect  of  superposed  plates  may  be  beautifully  shown  by  a  plate 
constructed  of  thin  mica  strips,  giving  approximately  a  half  wave  length 
retardation,  and  arranged  as  shown  in  Fig.  442,  partially  overlapping  in  paral- 
lel position,  crossed  at  90°,  and  crossed  at  some  other  angle. 

No  matter  what  maybe  the  inclination  of  the  vibration  directions  of 
the  crystal  to  those  of  the  nicol,  the  interference  color  will  remain  the  same 
since  the  retardation  remains  the  same.  A  rotation  of  the  stage  will  produce 
simply  a  change  in  the  intensity  of  the  color.  The  value  of  this  intensity, 
being  the  sum  of  the  intensities  of  all  the  emerging  rays,  may  be  represented  by 
an  equation  similar  in  form  to  that  of  equation  14,  Art.  285, 


(20) 


FIG.  442. — Arrangement  of  mica  strips 
to  show  the  effect  of  superposition  in 
different  positions. 


1=  2r*  cosV-  Sr2  sin  2(0-«*)-  sin  26-  sin2 


which  is  the  general  equation  for  the  intensity  of  the  emerging  ray  when 
white  light  is  used. 

With  nicols  crossed,  the  intensity  equation  is 


ART.  288J  EXAMINATION  BETWEEN  CROSSED  NICOLS  351 


With  parallel  nicols  it  is 

/-  Jr*  ±  Ir*  sin*  30  •  sin*  .  (22) 


With  two  superposed  mineral  plates  having  their  vibration  directions 
parallel,  and  their  slow  and  fast  rays  parallel  or  at  right  angles,  the  inter- 
ference colors  will  increase  or  decrease,  since  there  is  an  increase  or  decrease 
in  the  phasal  difference  of  the  combined  minerals,  wrhich  here  act  as  a  single 
mineral  section. 

In  the  above  it  must  be  remembered  that  throughout,  in  the  intensity 
equations,  the  same  values  for  the  refractive  indices  (HZ—  «i)  have  been 
assumed.  It  may  be  easily  seen  that  if  the  difference  between  the  indices 
increases,  the  value  of  the  intensity  increases,  and  vice  versa.  Now  in  aniso- 
tropic  media  the  maximum  difference  between  the  refractive  indices  occurs 
when  a  thin  section  is  cut  parallel  to  the  plane  of  the  optic  axes,  consequently, 
in  such  sections,  the  maximum  intensity  of  illumination  appears.  In  basal 
sections  of  uniaxial,  and  in  sections  cut  at  right  angles  to  the  optic  axes  in 
biaxial  minerals,  the  intensity  is  at  its  minimum. 

288.  Calculation  of  the  Value  of  the  Birefringence  in  any  Section.— 

Knowing  the  value  of  the  principal  re_frj£tive  indices  of  a  crystal  and  the  ori- 
entation of  the  section,  it  is  possible  to  compute  the  value  of  the  birefring- 
ence by  the  formula 

7'—  a=(y—  a)  sin  6  sin  6', 

where  0  and  0'  are  the  angles  between  the  normal  to  the  given  section  and 
the  two  optic  axes. 

In  1819,  Biot1  found  empirically  that  (7—  a)  sin  0  sin  0'  was  a  constant  for  all 
directions,  0  and  0'  being  the  angles  of  inclination  of  the  ray  to  the  optic  axes.  The 
mathematical  proof  for  this  proposition  was  given  by  Neumann2  in  1834. 

The  optical  ellipsoid,  in  the  most  general  case,  namely  in  a  biaxial  crystal,  is 
one  of  three  axes,  and  has  for  its  equation 

t2      -y2     z2 

^2+^2+72  =  i,  (Eq.  1,  Art.  63) 

or  aV2+  b2A'2+c222=i.  (Eq.  2,  Art.  63) 

Propagated  along  the  normal  to  the  section  there  will  be  two  waves  whose  veloc- 
ities are  inversely  proportional  to  7'  and  a,  the  major  and  minor  refractive  indices 

1  J.  B.  Biot:  Memoir?  sur  les  lots  generates  de  la  double  refraction  et  de  la  polarisation 
dans  les  corps  regulierement  cristallisees.      Mem.  Acad.  France.  Annee,  1818,  III  (1820), 
177-384,  especially  230. 

2  F.  E.  Neumann:  Ueber  die  optischen  Axen  und  die  Farben  zwei  axiger  Krystalle  im 
polarisirlen  Licht.     Pogg.  Ann.  XXXIII  (1834),  257-281. 


352  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  288 

in  the  section.  The  values  •/  and  a  will,  therefore,  represent  the  major  and  minor 
axes  of  the  section  of  the  ellipsoid  parallel  to  the  mineral  section  under  consideration. 
If  M  is  the  length  of  a  diameter  conjugate  to  the  other  two  (7  and  a),  and  p  is  the 
perpendicular  distance  between  the  planes  cf  the  section  and  the  tangent  to  the 
ellipsoid,  we  have  by  Cartesian  geometry: 


and  <-'*-j^- 

The  first  equation  may  be  written 

7"  +  «"      IT          111 


^' 

and  the  second 

y^  =  (^^)».  (4) 

Substituting  (4)  in  (3)  we  have 


.     •  (5) 

For  simplicity  in  writing,  let 


i  i 


Equation  (4)  becomes  ~j=Z,,  and  equation  (5) 


Subtracting  ^vm  from  each  member 

V*-21 

Extracting  the  square  root 
From  (9)  we  get 


(5a) 


~-  (6) 

From  (6)  we  have 

K2.  (7) 


-~'  (8) 


Now  the  normal  equation  of  a  plane  through  a  point  is 

X  cos  x+F  cos^-h  Z  cos  r  =/>.  (n) 


ART.  288]  EXAMINATION  BETWEEN  CROSSED  NICOLS  353 

But  the  plane  tangent  to  the  ellipsoid  Q2;t2-r-b2;y2-f-c2.32=  i  can  be  written 


in  which  XYZ  are  current  coordinates  and  xyz  those  of  the  point  of  contact. 
From  (u)  and  (12)  we  have 


cos  x  cos  i  cos  f 

=    ap'    ">'  =    bp'      C2=    c>'  (I3) 


/cosx\2      (cos  A2      /cosf\2  = 
V  a*p    I     h  V  V    /    h  \   c2/>  / 


/cos2   x  .  cos2  i     cos2  f 

(14) 
(15) 

(16) 

(i/) 

/>-    \        fl2                  b2     -            C2 

From  (13)  we  have 

Substituting  the  values  of  M2  and  pz  and  (15)  and  (14)  in  (sa), 
1  -+-  -  =  SL--S-' 

cos2  x    .cos2  i     cos2  f 
Rnf                            ^                                                                                                     * 

5      a2'    L2J  c2          cos2   x      cos2,     cos2  r 

a2             I2            c2 
cos2  x  /  i       i  \     cos2  i  /  i       i  \      cos2  f  /  i       i  \ 

a2     Vb'^c2/4     b2    \c2+a2/4       c2    \a2+b2/ 

COS2x        COS2  i       COS2  f 

a2            b2            c2 

>Z~abc/>' 

/>    < 

abc     A  /cos2  x  ,  cos2t  ,  cos2  f' 

cos2  t(c2+a2)+  cos2  r  (a2+b2).  (18) 


Substitute  the  value  of  (17)  in  (4) 


/  2^/2= ^2=  b2(;2  cos2  x+a2c2  cos2  t+a2b2  cos2  f-  (19) 

Equations  (18)  and  (19)  are  expressed  in  terms  of  cos  x,  cos  i,  and  cos  £.  They  may 
be  expressed  in  terms  of  directions  of  propagation  along  the  optic  axes.  In  a  man- 
ner similar  to  that  used  for  determining  the  value  of  tan2  V  in  Art.  71,  we  may  derive 
the  formulae  for  sine  and  cosine.  They  are 


cos2  V^- ^V-V  (2°) 

i       i      a2— c2 


23 


354  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  288 


?2     a2-b2 
sm<    K«__^=__.  (2Qfl) 

o^T2 

Between  two  directions  (cos  x,  cos  t,  cos  f)  and  (cos  x',  cos  /,  cos  f')  we  have 

cos  0  =  cos  x  cos  x'+cos  t  cos  t'+cos  f  cos  f'.  (21) 

When  cos  x'  =  sin  F,  cos  i'  =  o,  and  cos  $•'  =  cqs  F,  we  have 

cos  0  =cos  x  sin  F+cos  f  cos  F.  (22) 

When  cos  x/=  —sin  F,  cos  t'  =  o,  and  cos  f'  =  cos  F,  we  have 

cos  0'=  —cos  x  sin  F+cos  f  cos  F.  (23) 

We  also  have 

cos2  x  +cos2  t+cos2  r=  i.  (23a) 

Solving  Eq.  (22)  and  (23)  for  cos  x,  cos  t,  and  cos  f,  and  squaring,  we  have 

cos  0 — cos  0r 


and 


cos  = 


cos    f. 


., 

2  COS    F 

Squaring  Eq.  (24)  and  (25)  and  combining  with  Eq.  (20), 

)2      a2-c 


2  ._  2-2 

^ 

(cos  «-  cos  0')2      a2-c2 

cosx=     ~4~      '*=*' 

Substituting  values  from  Eq.  (27),  (233),  and  (26)  in  (18),  we  obtain 

42+42  -  »2+  «•-  (^L?^)  V-  a+^tpilV-  [2) 

=  a2+  c2+  (a2  -  c2)  cos  0  cos  e'.  (28) 

Substituting  the  same  values  in  (19)  we  have, 


^7^  =  cos2xb2c2+(i-cos2  x-cos2  r)a2c2+cos2ra2b2 

=  a2c2+(b2-a2)c2  cos2  x+(b2-c2)  a2  cos2  r 
=  a2c2-  ^Q2_  b2y(a2-c2)(cos  0-cos  O2+^^7c^-(a2-c2)(cos  0+cos  0')2 

=  a2c2-   -  --  -  c2(cos  0-cos  0')2+       — a2(cos  0+cos  0')2 

=  a2c2_a2c2-c4  _  a4-q2c2 

4  4 

=  a2c2-  --(cos2  0-2  cos  0  cos  0/+cos2  0')+  (^-°2^- j  (cos2  0+2  cos  0  cos  0' 

\       4      / 

+  COS2  0') 


ART.  289]  EXAMINATION  BETWEEN  CROSSED  NICOLS  355 

r/c4     a2c2\,  /c4-o2c2\  ,1 

=  a2c2+    ( )  (cos2  0+cos-  0  )—  (  —        — )  cos  0  cos  0 

I-  \4         4  \        2        /  -I 

'cos2  0+cos2  0')+  (  -         — )  cos  0  cos 


[  (--—  )  ( 


=  a2c2+  (COS2  ,+cos2  0+  cos  e  cos  0' 

=  aaca+-  --r—  (cos*  0+cos2  *0-f  ^-^-cas  0  cos  0'.  (29) 

4  2 

Again,  substituting  in  equation  (10),  we  have 
(~-^J=(a2+c2)2+(a2-c2)2(cos2  e  cos2  0'-cos2  0-cos2  0')-4a2c2.      (30) 

\a  i     7  V 

From  trigonometry  we  have 

(i—  cos2  0)(i—  cos2  0')  =  i—  cos2  8  —  cos2  0r+cos2  6  cos2  6', 
and  this  equation  substituted  in  (30)  gives 


=(a2+c2)2+(a2-c2)[(i-cos2  0)(i-cos2  e'^-i}-^^ 

=  (a2+c2)2+(a2-c2)2(sin2  e  sin2  0'-i)-4a2c2 
=  (a2-c2)2  sin2  0-sin2  e'. 
Extracting  the  square  root  we  have 

-4^)  =  (a2-  c2)sin  e  sin  0'.     .  (31) 

But  a  =  ~~,  and  c  =  -.  whereby 

a  y' 

I        I       /I      I  \   . 
-2-i=        ~       sm  tf  sm  e'  (32) 


Since  the  value  of  the  birefringence  7  —  a  is  generally  small,  we  may  write  with 
approximate  accuracy 

7'  —  a  =(7  —  a)  sin  0  sin  0r.  (33) 

This  is  the  desired  equation  for  calculating  the  value  of  the  birefringence  of  any 
section. 

289.  Lines  of  Equal  Birefringence.  —  Curves  of  equal  birefringence  were 
first  used  by  Michel-Levy.1  They  may  be  readily  determined  by  the  equa- 
tion 7'  —  a'  =  (7  —  a)  sin  6  sin  0',2  and  the  results  may  be  plotted  by  'tracing 
the  curve  produced  by  the  poles  of  the  given  sections.  Such  curves  are  of 
value  in  showing  the  relative  accuracy  of  random  sections  of  a  mineral  in 
comparison  with  sections  of  known  orientation  whose  properties  are  known. 

1  A.  Michel-Levy:  Etude  sur  la  determination  des  jeldspaths.  Premiere  fascicule, 
Paris,  1894. 

-  Art.  288,  Eq.  33. 


356  MANUAL  OF  PETROGRAPHIC  METHODS 

The  equation  may  be  written 


[ART.  289 


7'  -a1 
7  —  a. 


=  sin  6  sin  6'. 


The  desired  curve  is  represented  by  a  given  value  of  birefringence  (7'  —  *r ) 


K=  0.10     0.20      0.30       0.40       0.50        0.60  0.70  0.80 


0=0  10J  20°  30°  40s  50°  60'  70'  80'  90' 

PIG.  443. — Diagram  for  computing  the  percentage  of  the  maximum  birefringence  which  appears  in 

any  section.     (After  Wright.1) 

which  is  therefore  a  constant,  as  is  also  7-0:,  for  any  given  mineral.     The 
equation  therefore  becomes 

K  =  sin  0  sin  9' 

in  which  K  is  less  than  unity,  since  7— <*  represents  the  maximum  bire- 
fringence. 

In  Fig.  443  the  abscissae  and  ordinates  represent  values  of  0  and  6',  and 


7  —  ex. 
the  curves  the  birefringence  ratio  K  =      _a  f or  values  of  o.i,  0.2,0.3,  .   .  . 

1  Duparc  and  Pearce,  in  their  Traile  de  technique  miner alogique  el  petrographique,  I, 
Leipzig,  1907,  229,  give  a  similar  diagram  but  use  equal  spaces  for  the  sines  of  6  and  6' 
instead  of  equal  spaces  for  the  angles  themselves. 


ART.  289]  EXAMINATION  BETWEEN  CROSSED  NICOLS  357 

i  .o.  The  curves  are  equilateral  hyperbolae  whose  crests  lie  on  a  line  making 
an  angle  of  45°  with  the  coordinates. 

By  the  use  of  this  diagram  the  ratio  of  the  birefringence  of  almost  any 
section  to  the  maximum  birefringence  may  be  determined  if  the  angle  between 
the  normal  to  the  section  and  the  optic  axes  is  known.  An  exception  occurs 
when  the  section  lies  within  the  zone  of  circles  tangent  internally  and  exter- 
nally. This  is  best  shown  by  constructing  curves,  in  stereographic  projection, 
through  the  poles  having  the  same  percentage  of  the  maximum  birefringence, 
for  while  the  stereographic  projection  is  somewhat  distorted  toward  the  mar- 
gin, the  drawing  will  give  a  general  idea  of  the  actual  appearance  of  the  curves. 

If  to  K,  in  the  equation  above,  there  be  given  definite  values,  such  as  o.i, 
0.2, 0.3,  etc.,  and  there  be  assigned  to  6  various  values  ranging  from  o°  to  90°, 
the  corresponding  values  for  0'  may  be  determined,  or  they  may  be  taken 
directly  from  the  diagram  in  Fig.  443,  and  from  these  values  the  curve  may 
be  plotted  in  stereographic  projection.  The  method  is  as  follows:  Locate 
the  optic  axes  in  the  projection,  and  with  these  as  centers  draw  circles  of 
proper  radii  from  each.  Thus,  for  K  =  o.i  we  have  the  following  values 

5  =  90°  0'  =  5.8° 

73-5  6.0 

S4-o  7-0 

45-5  8.0 

39-5  9-o 

35 .o  10. o 

29.0  12.0 
20. o                      .        17  .o 

17.0  20. o 

12. o  29.0 

10. o  35.0 

9-0  39-5 

80  45.5                                                FIG.  444. 

7-0  54-0 

6.0  73.5 

5.8  90.0 

With  the  optic  axis  A  (Fig.  444)  as  a  center,1  draw  a  circle  of  90°  radius 
(0),  and  with  B  as  a  center  a  circle  of  5.8°  (0')-  The  intersections  of  the  two 
will  give  two  points  in  every  case  except  where  the  circles  are  tangent  inter- 
nally or  externally,  in  which  case  only  one  point  will  occur  (c  or  d,  Fig.  444). 
Proceed  likewise  for  0  =  73°,  0'  =  6°,  and  so  on,  until  enough  points  have  been 
obtained  to  trace  the  curve.  In  a  similar  manner  proceed  with  K  =  o.2,  and 
so  on. 

The  curves  are  determined  much  more  easily  by  means  of  a  stereographic 

1  It  must  be  borne  in  mind  that  when  one  speaks  of  drawing  a  circle  in  stereographic 
projection,  about  a  point  as  a  center,  that  a  point  on  the  sphere  is  meant.  The  curves, 
although  true  circles,  will  not  be  concentric  in  the  projection,  although  actually  so  on  the 
sphere. 


358 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  289 


net,  as  was  pointed  out  by  Wulff.1     He  prepared  a  table,  once  for  all,  for 

y'-a 
various  values  of  the  birefringence  ratio  — —  -  (Column  I)  and  definite  values 


7'  —  a' 

90 

88 

86 

84 

82 

80 

78 

76 

74 

72 

70 

68 

66 

64 

62 

60 

58 

56 

54 

52 

50 

48 

46 

44 

42 

40 

38 

36 

34 

32 

30 

28 

26 

24 

„' 

y  —  a 

O.  IO 

6 

6 

6 

6 

'6 

6 

6 

6 

6 

6 

6 

6 

6 

6 

8 

8 

8 

8 

9 

9 

9 

IO 

IO 

II 

12 

„ 

13 

14 

" 

o.  20 
0.30 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

12 

13 

13 

13 

13 

14 

14 

14 

15 

15 

15 

16 

17 

17 

18 

19 

20 

21 

22 

24 

25 

18 

18 

18 

18 

18 

1  8 

18 

18 

18 

18 

19 

19 

19 

19 

20 

20 

21 

21 

22 

22 

23 

24 

25 

26 

2? 

28 

29 

31 

32 

0.40 

24 
30 

-'4 

24 

24 

24 

24 

24 

24 

24 

25 

25 

26 

26 

26 

2? 

27 

28 

20 

30 

31 

31 

32 

34 

35 

37 

38 

0.50 

0.60 
0.70 

30 

30 

30 

30 

31 

31 

31 

31 

32 

32 

33 

33 

34 

34 

35 

36 

37 

38 

39 

41 

42 

44 

37 

37 

37 

37 

37 

38 

38 

38 

39 

39 

40 

40 

4\ 

4* 

43 

44 

45 

46 

48 

50 

44 

44 

45 

45 

45 

45 

46 

46 

47 

47 

48 

49 

50 

51 

52 

54 

56 

0.75 
0.80 

49 

49 

49 

49 

49 

SO 

50 

51 

51 

52 

53 

54 

55 

56 

58 

60 

S3 

53 

53 

53 

54 

54 

55 

55 

56 

57 

58 

60 

61 

63 

0.85 

58 

58 

58 

59 

59 

60 

60 

61 

62 

63 

65^66 

0.90 

64 

64 

6465 

65 

66 

67 

68 

69 

71 

0.95 

72 

72 

72 

73 

74 

75 

77 

of  a.  The  values  for  a ',  in  even  degrees,  are  given  at  the  intersections  of 
horizontal  and  vertical  lines  through  these  two  values. 

Further,  once  for  all,  upon  a  piece  of  tracing  paper,  the  vertical  small 
circles  of  half  the  Wulff  net  are  drawn  from  one  pole  to  the  equator.  This 
tracing  is  placed  concentrically  over  a  Wulff  net  (Fig.  32)  in  such  a  position 
that  its  pole  lies  on  the  periphery  at  a  distance  of  the  true  axial  angle  (2V) 
from  the  pole  of  the  net,  and  is  fastened  in  this  position  by  means  of  thumb 
tacks.  There  will  now  appear,  in  the  desired  quadrant,  a  double  net  of 
vertical  small  circles  whose  intersections  will  give  the  angles  a  and  a'  from 
the  points  of  emergence  of  the  optic  axes  on  the  periphery  of  the  circle;  By 
placing  above  the  double  net  a  clean  sheet  of  tracing  paper,  the  desired  curves 
may  be  drawn  through  the  proper  intersections  as  given  in  the  above  table. 
In  this  way  the  curves  of  equal  double  refraction  are  projected,  for  a  single 
quadrant,  in  a  plane  parallel  to  the  plane  of  the  optic  axes.  The  other  three 
quadrants  are  symmetrical  with  the  first,  and  may  be  reproduced  by  tracing 
it  (Fig.  445).  The  true  values  of  the  curves  may  be  obtained  by  multiplying 
their  ratio  values,  as  given  in  the  table,  by  the  value  of  the  maximum  double 
refraction  of  the  mineral  in  question.  The  projection  plane  may  be  changed 
to  any  other  plane  desired  by  the  method  given  in  Art.  16,  problem  9. 

The  curves  in  Figs.  445  and  446,  which  are  those  of  albite  (AbgsAnz, 
with  27  =  77°),  were  constructed  in  this  manner.  Analyzed  independently, 
the  fringes  form  closed  circles  when  #<sin2  V  (Fig.  444,  curve  0.2):  when 

1  Georg  Wulff:    Untersuchungen  im  Gebiete  der  optischen  Eigenschaften  isomorpher  Krys- 
lalle.     Zeitschr.  f.  Kryst.,  XXXVI  (1901),  20-22. 


ART.  290] 


EXAMINATION  BETWEEN  CROSSED  NICOLS 


359 


K  =  sin2  V  they  become  tangent  at  the  center  over  the  acute  bisectrix  and 
form  a  "  figure  eight."  When  sin2  F>A'<cos2  V  the  curves  do  not  pass 
between  the  optic  axes  in  the  acute  axial  angle,  but  form  a  closed  curve 
around  the  two  axes.  When  K  =  cos2  V,  the  curve  becomes  tangent  at  the 
obtuse  bisectrix;  it  still  forms  a  closed  curve  and  extends  around  both  optic 
axes.  When  K>cos2  V  the  circles  do  not  become  tangent  and  the  curves 
are  open. 


FIG.  445. — Lines  of  equal  birefringence  in  a 
section  of  albite  cut  parallel  to  the  plane  of  the 
optic  axes.  (After  Wulff.) 


FIG.  446. — Lines  of  equal  birefringence 
in  a  section  of  albite  cut  parallel  to  ooi. 
(After  Wulff.) 


From  these  different  cases  it  is  seen  that  when  K  <sin2  V,  for  certain  values 
of  0,  the  curves  cross  the  plane  of  the  optic  axes  in  both  the  acute  and  the 
obtuse  optic  axial  angles;  K  =  sin2  V  is  the  limiting  case.  When  K<  cos2  V 
but  >  sin2  V  the  curves  cross  the  plane  of  the  optic  axes  in  the  obtuse  optic 
axial  angle,  and  K  =  cos2  V  is  the  limiting  case.  The  minimum  birefringence 
is  given  by  sections  cut  at  right  angles  to  an  optic  axis,  and  the  maximum 
birefringence  by  sections  parallel  to  the  plane  of  the  optic  axes. 

290.  Abnormal  Birefringence. — Owing  to  the  fact  that  the  retardation 
of  rays  of  different  colors  is  not  exactly  the  same  for  all,  the  resulting  inter- 
ference color  is  not  the  pure  complementary  color  of  that  extinguished.  This 
may  be  seen  by  inspecting  the  following  table  of  retardations  in  calcite  for 
various  colors  of  the  spectrum. 


Color 

Fraunhofer 
line 

Wave 
length 

CO 

•  € 

OJ-€ 

Red.. 

A 

7  ^0   4O 

i  6>o 

I     4.83 

o  167 

Red... 

B 

686  74 

i   6^3 

I     484 

o  169 

Yellow  
Blue  
Violet  .  .  . 

D 
F 
H 

589.60 
486.15 

306  81 

1.658 

1.668 
i  681 

1.486 
I.4QI 

I          4.O8 

o.  172 

0.177 

o  i8«; 

360  MANUAL  OF  PETROGRAPHIC  MP:THODS  [ART.  290 

The  double  refraction  for  the  color  at  one  end  of  the  spectrum  is  con- 
siderably less  than  that  at  the  other  and,  as  a  consequence,  the  resulting  inter- 
ference color  will  not  be  normal.  When  the  double  refraction  for  the  red 
is  less  than  that  for  the  violet,  as  it  is  in  calcite,  Becke  called  this  abnormal 
color  supernormal  (ubernormal);  when  the  reverse  is  the  case,  subnormal 
(unter  normal). 

Another  cause  of  abnormal  colors  is  the  fact  that  in  certain  minerals  the 
birefringence  is  zero  for  certain  wave  lengths.  For  example,  fuggerite1 
(Ca3Al2Si2Oio),  at  one  end  of  the  spectrum,  is  positive  (e>  w),  at  the  other, 
negative  ( o>  >  e) ,  and  for  sodium  light  it  is  dark  (u  =  c) .  By  white  light,  owing 
to  the  extinguishing  of  the  yellow;  the  complementary  color,  deep  blue,  ap- 
pears. The  same  color,  naturally,  will  appear  no  matter  what  may  be  the 
thickness  of  the  slide,  the  color  being  only  deeper  in  thick  sections.  That  the 
color  is  abnormal  may  be  seen  by  inserting  a  Johannsen  wedge,  or  any  other 
accessory  plate  which  will  compensate  for  less  than  1/4  wave  length.  Instead 
of  producing  darkness,  as  it  ordinarily  would  at  the  point  of  compensation, 
a  pale  brownish  color  appears.  The  blue  of  the  second  order,  which  some- 
what resembles  the  abnormal  color,  is  reduced  to  a  bright  orange  by  the 
same  retardation. 

The  abnormal  blue  color  is  likewise  shown  by  melilite,  vesuvianite,2 
and  chlorite.  Certain  other  minerals  show  different  abnormal  colors,  de- 
pending upon  the  wave  length  which  is  totally  extinguished  in  them. 

Another  cause  for  abnormal  interference  colors  is  the  dispersion  of  the 
directions  of  vibration  in  monoclinic  and  triclinic  crystals.  Since  all  of  the 
colors  are  not  extinguished  at  the  same  time,  an  abnormal  color  results.  Zoi- 
site  is  an  example. 

A  third  cause  for  abnormal  colors  may  be  found  in  the  fact  that  part  of 
the  light  from  an  illuminating  system  of  large  aperture  does  not  pass  through 
the  crystal  in  strictly  parallel  directions,  but  at  an  angle.  Traveling  thus 
different  paths,  the  amount  of  the  retardation  will  be  different,  and  a  mixed 
interference  color  results. 

A  fourth  cause  for  abnormal  colors  is  the  modification  produced  by  the 
color  of  the  mineral  itself.  Thus  chlorite,  when  of  a  deep  green,  may  show 
a  greenish  interference  color  instead  of  the  normal  blue,  and  that  of  biotite 
or  hornblende  may  appear  to  be  that  of  the  mineral  itself. 

1  E.  Weinschenk:    Fuggerit,  ein  neues  Mineral  aus  dem  Fassathal.     Zeitschr.  f.  Kryst., 
XXVII  (1896-7),  577-582. 

2  C.  Hlawatsch:    Beslimmung  der  Doppelbrechung  fur  verschiedene  Farben  an   einigen 
Miner  alien.     T.  M.  P.  M.,  XXI  (1902),  107-156. 


CHAPTER  XXIV 

DETERMINATION  OF  THE  VIBRATION  DIRECTIONS  IN  MINERAL 

PLATES 

291.  Optical  Character  of  the  Elongation. — It  has  already  been  pointed 
out1  that  when  two  anisotropic  minerals  are  superposed,  the  resulting  inter- 
ference color  is  their  algebraic  sum.  This  principle  is  made  use  of  in  deter- 
mining the  directions  of  the  fast  and  slow  rays  in  mineral  plates. 

If  an  anisotropic  mineral  plate,  in  which  the  vibration  directions  are 
unknown,  is  placed  upon  the  stage  of  the  microscope  between  crossed  nicols, 
and  it  is  rotated  until  no  light  is  transmitted,  its  vibration  directions  lie 
parallel  to  those  of  the  polarizer  and  analyzer.2  If  it  is  rotated  still  farther, 
until  its  vibration  directions  make  angles  of  45°  with  its  former  position,  it  will 
be  in  the  position  of  maximum  illumination.  If,  now,  there  is  placed  above  it, 
also  in  its  position  of  maximum  illumination,  a  mineral  plate  in  which  the 
vibration  directions  are  known,  it  may  be  seen,  readily,  that  if  the  interference 
color  rises  in  the  scale,  the  vibration  directions  of  the  unknown  mineral  are 
parallel  to  those  of  the  known,  and  if  it  sinks,  at  right  angles.  Various  min- 
eral plates,  with  the  vibration  direction  of  the  slow  ray  (usually)  marked 
by  arrows,3  are  provided  with  petrographic  microscopes.  The  most  common 
are  the  quarter- wave  plate,  gypsum  plate,  and  quartz  wedge. 

The  determination  of  the  fast  and  slow  rays  in  a  crystal  section  may  or 
may  not  be  of  value  in  its  determination.  If  the  orientation,  that  is  the  rela- 
tion of  the  vibration  directions  to  crystallographic  directions,  is  known  in 
uniaxial  crystals,  the  determination  of  the  fast  or  slow  ray  determines  the 
optical  character  of  the  mineral  itself,  for  if  crystallographic  c  is  the  fast  ray, 
the  mineral  is  negative,  if  it  is  the  slow  ray,  positive.4  In  biaxial  crystals  the 
optical  character  of  the  mineral  may  be  determined  by  the  orientation  in 
any  section  if  the  positions  of  the  optic  axes  are  known.5 

But  crystals  have  characteristic  cleavages,  consequently  the  fragments 
found  in  rock  sections  are  commonly  bounded  by  cleavage  planes.  The 

1  Art.  286  supra. 

2  Art.  283  and  Art.  285,  Case  II,  supra. 

3  The  arrow  so  marked  — >c,  does  not  mean  that  the  slow  ray  travels  to  the 

right,  but  that  its  plane  of  vibration  is  parallel  to  the  shaft,  thus  < —       — >.     This  explana- 
tion may  seem  absurdly  unnecessary,  but  apparently  is  not,  judging  from  questions  asked 
by  students. 

•Art.  51. 
6  Art.  75- 

361 


362  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  292 

determination  of  the  vibration  directions  in  such  pieces  may  separate  two 
similar  minerals,  for  in  one  the  fast  ray  may  be  parallel  to  the  cleavage 
and  in  the  other  at  right  angles  to  it.  Minerals  in  which  such  cleavages  are 
common  usually  occur  in  lath-like  fragments  in  rock  sections,  and  this 
characteristic  extension  in  one  direction  is  spoken  of  as  the  elongation  of  the 
mineral  and  includes  both  cleavage  elongation  and  prismatic  elongation  of 
natural  crystals.  If  the  fast  ray  coincides  with  the  long  direction,  the  elonga- 
tion is  said  to  be  negative,  if  the  slow  ray  coincides,  positive.1 

ACCESSORIES  USED  FOR  THE  DETERMINATION  or  THE  VIBRATION  DIRECTIONS 

OF  A  MINERAL 

292.  Kinds  of  Accessories. — Only  the  simplest  forms  of  the  many  acces- 
sories which  may  be  used  for  the  determination  of  vibration  directions  are 
given  below.     Many  others2  are  described  under  the  methods  for  the  deter- 
mination of  birefringence  and  extinction  angles. 

The  simple  forms  may  be  grouped  into  two  classes. 

I.  Simple  plane-parallel  plates. 

a.  Quarter  undulation  mica  plate. 

b.  Unit  retardation  plate. 

II.  Wedges. 

a.  Simple  quartz  or  gypsum  wedge. 

b.  Fedorow  mica  comparator. 

c.  Wright  combination  wedge. 

d.  Johannsen  quartz-mica  wedge. 

I.     SIMPLE  PLANE  PARALLEL  PLATES 

293.  Quarter  Undulation  Mica  Plate.— The  quarter  undulation  mica 
plate,  also  called  quarter  wave-,  quarter  order  mica-,  or  1/4  X  plate,  is  made 
of  such  a  thickness  that  M(n2— «i)  =  i/4  X,3  whereby  one  vibration  will  be 
retarded  a  quarter  of  a  wave  length  behind  the  other,  and  the  transmitted 
wave  will  be  elliptically  polarized. 

The  fact  that  two  mineral  sections  superposed  at  right  angles  to  each  other 
show  a  reduction  in  the  interference  tint,  was  discovered  by  Arago4  in  1811, 

1  The  signs  of  the  optical  character  of  the  elongation  and  of  the  optical  character  of  the 
mineral  may  be  remembered  by  connecting  them  thus:  When  the  c  axis  of  a  uniaxial  or 
the  acute  bisectrix  of  a  biaxial  mineral  is  the  fast  ray,  the  mineral  is  negative;  when  the 
elongation  is  parallel  to  the  fast  ray,  the  elongation  is  negative. 

2  Various  accessories  are  also  described  in: 

G.  Valentin:  Die  Untersuchungen  der  Pflanzen-  und  der  Thiergeivebe   im  polarisirten 
Lichte.     1 86 1.* 

Moigno:  Repertoire  d'optique  moderne,  1850,  Tome  IV,  1592  et  seq.* 

3  Equation  3,  Art.  280,  supra. 

4  F.  Arago:  Memoir e  sur  une  modification  remarquable  qu'eprouvent  les  rayons  lumineux 
dans  leur  passage  d  travers  certains  corps  diaphanes,  et  sur   uelques  autres  nouveaux  phenomenes 
d'optique.     Mem.  Acad.  France.  Annee  1811,  Pt.  I.  XII  (1812),  93-134. 


ART.  293]  VIBRATION  DIRECTIONS  IN  MINERAL  PLATES  363 

but  Biot1  was  the  first  to  suggest  that  the  vibration  directions  in  an  unknown 
mineral  could  be  determined  by  comparison  with  those  of  one  which  is  known. 
He  used  thin  plates  of  mica,  gypsum,  quartz,  and  other  substances  whose 
vibration  directions  were  determined.  He  did  not  specify  any  particular 
thickness  of  plate,  but  had  a  series  of  different  thicknesses,  and  chose  which- 
ever compensated  with  the  unknown  mineral.  If  the  mineral  under  examina- 
tion had  the  same  optical  character  as  the  plate,  he  called  it  attractive,  if 
the  opposite,  repulsive.  To  such  minerals  Brewster2  gave  the  names  positive 
and  negative. 

The  use  of  thin  plates  of  definite  thicknesses  was  probably  introduced  by 
Airy3  in  1831.  He  showed  that  light  would  be  circularly  polarized  by  plates 
of  1/4,  3/4^  5/4,  etc.,  retardation.  De  Senarmont,4  in  1851,  first  applied  a 
half- wave  plate  to  the  determination  of  the  three  vibration  axes  of  crystals. 
A  quarter  undulation  plate  was  used  by  Bravais,5  in  1855,  and  since  then  it 
has  been  in  common  use  as  an  accessory  in  petrographic  microscopic  work. 

The  most  convenient  mineral  from  wrhich  a  quarter  undulation  plate  can 
be  constructed  is  muscovite.  It  cleaves  in  plates  which  may  be  made  of 
almost  any  degree  of  thinness,  and,  since  these  plates  differ  but  2°  from  being 
perpendicular  to  the  acute  bisectrix  of  the  optic  axial  angle,  this  bisectrix 
emerges  in  the  center  of  the  field.  Since  muscovite  is  negative,  the  bisec- 
trix is  the  fast  ray  a.  The  other  vibration  directions  may  be  determined  by 
examining  the  mineral  plate  in  convergent  polarized  light.6  In  the  inter- 
ference figure  thus  obtained,  the  slowest  ray  c  will  vibrate  in  the  direction 
of  the  line  connecting  the  points  of  emergence  of  the  optic  axes,  that  is,  the 
points  of  rotation  of  the  black  bars;  b  is  the  direction  at  right  angles  to 
c  and  also  lies  in  the  cleavage  flake. 

1  J.  B.  Biot:    Memoire  sur  line  noui'elle  application  de  la  theorie  des  oscillations  de  la 
lumicre.    Lu  a  1'Institute  27  dec.  1813.     Mem.  Acad.  France,  Annee   1812,  XIII,  Paris, 
1816,  Pt.  II,  1-18. 

Idem:  Traite  de  physique.     Paris,  1816,  IV,  420-422,  543-566. 

2  David  Brewster:    On  the  laws  of  polarization  and  double  refraction  in  regularly  crystal- 
lized bodies.     Phil.  Trans.  Roy.  Soc.  London,  CVIII  (1818),  199-273,  in  particular  219. 

3  G.  B.  Airy:    On  tJte  nature  of  the  light  in  the  two  rays  produced  by  tlie  double  refraction  of 
quartz.     Read  Feb.  21,  1831.     Cambridge  Phil.  See.,  IV  (1833),  79-123. 

Idem:  Addition  to  a  paper  "On  the  nature  of  the  light  in  the  two  rays  produced  by  the 
double  refraction  of  quartz."  Read  April  18,  1831.  Cambridge  Phil.  Soc.,  IV  (1833), 
199-208. 

Idem:  Ueber  die  Natur  des  Lichtes  in  den  beiden  durch  die  Doppelbrechung  des  Berg- 
krystalls  hen-orgebrachten  Strahlen.  Pogg.  Ann.,  XXIII  (1831),  204-280. 

4  H.  de  Senarmont:    Recher cites  sur  les  proprietes  optiques  birefringentes   des  corps  iso- 
morphes.   .Ann.  d.  chim.  et  phys.,  XXXIII  (1851),  391-401. 

5  A.  Bravais:    Beschreibung  eines  neuen  Polariskops  und  Untersuchung  iiber  die  schwachen 
Doppelbrechungen.     Pogg.  Ann.,  XCVI  (1855),  395-414. 

Idem:  D'un  nouveau  polariscope  et  recherches  sur  les  doubles  refractions  peu  energiques. 
Ann.  d.  chim.  et  phys.,  XLIII  (1855),  129-149. 

6  See  Chapter  XXIX. 


364  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  293 

To  prepare  a  quarter  undulation  mica  plate,  select  a  clear  piece  of  mus- 
covite  and  split  it  into  very  thin  plates  by  inserting  a  pin  between  the  lamellae. 
It  is  not  possible  to  obtain  large  lamellae  uniformly  thin  throughout,  and  it  is 
therefore  advisable  to  examine  them  between  crossed  nicols  and  scratch 
lines  upon  tne  surface  around  the  areas  of  like  interference  colors.  The 
cleavage  plate  may  then  be  cut  apart  on  the  contour  lines  and  pieces  of  like 
thickness  kept  separate  for  different  purposes.  For  the  quarter  undulation 
plate  select  such  pieces  as  give  a  pale  neutral  gray  interference  color,  and  whose 
first  interference  ring  makes  a  perfect  ellipse  around  both  the  eyes.1  They 
may  be  tested  further  by  comparison,  by  compensation,  with  a  standard  i/4\ 
plate.  They  should  be  of  such  thickness  that  by  sodium  light  the  retardation 
is  just  a  quarter  of  a  wave  length. 

From  plates  so  prepared,  rectangular  pieces  should  be 
cut,  parallel  or  at  right  angles  to  the  vibration  direction 
of  the  slow  ray,  the  directions  being  determined  by  exami- 
nation of  the  interference  figure.  The  films,  finally,  should 
be  mounted  in  Canada  balsam  between  glass  plates.  The 
vibration  direction  of  the  slow  ray  should  be  indicated  by 
an  arrow  scratched  on  the  glass.  As  usually  mounted, 
the  c  direction  lies  directly  across  the  slip  (Fig.  447).  In 

FIG.  447.  —  Quarter  un-  .  ,,  ,    ,     ,.         .,  .  ,,    •, 

duiation  mica  plate.    some  microscopes  the  slot  for  the  accessories  is  parallel 

to  the  cross-hairs,  therefore  the  c  direction  of  the  mica 

must  lie  at  45°  with  the  long  direction.     When  cut  in  the  former  of  the  two 

way*,,  it  is  easy  to  remember  that  the  slow  vibration  is  parallel  to  the  short, 

and  the  fast  vibration  parallel  to  the  long  edge  of  the  plate. 

The  thickness  of  a  mica  plate  necessary  to  produce  a  retardation  of  i/4\  may  be 
computed  from  equation  4,  Art.  280, 


Since  the  section  is  cut  at  right  angles  to  the  acute  bisectrix  (a),  it  contains  the 
axes  b  and  c  with  indices  n2  and  n\t  equal  to  7  and  0.  If  these  are  1.603  and  1.595 
in  the  specimen  of  mica  used,  we  have,  for  sodium  light, 

i  =  M  (1.603-1.595) 

4  ~  0.000589 

whereby 

,,      0.000589 

M  =  -          -~  =  0.0184  mm. 
4X0.008 

To  determine  the  fast  and  slow  vibration  directions,  the  process  is  as  given 
above.  Turn  the  mineral,  between  crossed  nicols,  45°  off  the  position  of  dark- 
ness, and  insert  the  mica  plate  in  the  slot  provided  for  it.  If  the  interference 
color  of  the  mineral  is  increased  by  1/4  X  the  slow  rays  of  the  two  are  parallel, 
if  it  decreases,  they  are  at  right  angles. 

Further  uses  for  the  mica  plate  are  given  in  Art.  404. 

1  See  Art.  360  and  Fig.  561. 


ART.  295]          VIBRATION  DIRECTIONS  IN  MINERAL  PLATES 


365 


2Q4-  Unit  Retardation  Plate. — A  plate  whose  retardation  is  equal  to  5  7  5 MM, 
the  wave  length  of  rays  near  the  D  line,  extinguishes  the  intense  yellow  rays, 
and  the  resulting  color  is  the  sensitive  violet.1  If  such  a  plate  is  cut  from 
mica  it  must  be  four  times  as  thick  as  the  quarter  undulation  plate  just  de- 
scribed. Since  mica  is  rarely  entirely  colorless,  such  a  plate  generally  has  a 
yellow  tinge,  and  for  that  reason  gypsum  usually  is,  though  quartz  may  be, 
used.  Gypsum  is  monoclinic,  the  angle  /?  =  8o°42r,  b=  b,  and  c  10  =  53° 
in  the  obtuse  angle  (Fig.  448),  oio  cleavage  good, 
in  and  loo  distinct.  In  the  oio  cleavage  flakes 
lies  the  plane  of  the  optic  axes,  and  the  b  direction 
is  perpendicular  to  it.  The  first  order  violet  plate 
may  be  cut  with  its  long  direction  parallel  either  to 
a  or  c.  To  avoid  confusion  it  is  better  that  the 
elongation  of  all  of  the  accessories  be  the  same.  The 
retardation  corresponds  practically  to  a  wave  length 
of  the  mean  of  white  light,  and  the  plate  may  there- 
fore be  spoken  of  as  the  unit  retardation  plate.  It 
is  usually  called  the  Violet  of  the  first  order,  "Red" 
of  the  first  order,  or  Sensitive  plate. 

The  thickness  may  be  calculated  as  before. 

0.00x5589 
If  7  — «  =  o.oo95,  \  =  M(y—  a),  M  =  OOQ. =  0.062  mm. 

If  quartz  is  used,  it  should  be  cut  to  a  plane  parallel  to  the  optic  axis  and  of  a 
.000589  _ 
.009 


FlG-  448.— Orientation  of  the 
unit  retardation  PIate- 


thickness  M  = 


mm. 


The  first  use  made  of  a  unit  retardation  plate  seems  to  have  been  by 
Biot2  in  1813  and  the  term  teinte  sensible  was  introduced  by  him. 

For  the  determination  of  extinction  angles,  slight  double  refractions,  and 
the  optical  character  of  minerals  by  means  of  this  plate,  see  Arts.  319,  334, 
and  405.  For  the  determination  of  vibration  directions  in  mineral  plates,  the 
method  is  exactly  the  same  as  that  just  described  for  the  mica  plate. 


II.  RETARDATION  WEDGES 

295.  Simple  Quartz  or  Gypsum  Wedge. — Instead  of  using  a  plane-parallel 
plate,  it  is  often  convenient  to  use  one  of  a  wedge-shape.  When  such  a  plate 
is  inserted  in  the  microscope  between  crossed  nicols,  it  will  not  be  uniformly 
colored,  but  will  show  parallel  bands  (Fig.  449)  corresponding  to  the  whole 

1  Teinte  sensible  of  Biot,  teinte  de  passage. 

2  ].  B.  Biot:  Memoire  stir  nne  now  die  application  de   la  theorie  des  oscil'ations  de  la 
lumiere.    Lu  a  1'Institute  27  dec.,  1813.     Mem.  Acad.  France.  Annee  1812,  Paris,  XIII 
(1816),  pt.  ii,  1-18. 

Idem:  Trails  de  physique,  Paris,  1816,  IV,  420-422,  543-566. 


366  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  296 

range  of  Newton's  colors,  and  varying  from  nearly  darkness  to  colors  of  the 
third,  fourth,  or  higher  orders.  The  explanation  is,  of  course,  that  the 
increased  thickness  of  wedge,  as  it  is  shoved  into  the  field,  causes  more  and 
more  retardation  of  the  transmitted  rays.  When  such  an  accessory  is  in- 
serted, with  its  thin  edge  foremost,  above  a  mineral  plate,  there  will  be  a 
gradual  reduction  of  the  interference  colors  to  zero  when  the  vibration  direc- 
tions of  the  two  are  at  right  angles.  At  the  point  where  the  retardations  of 
the  two  are  equal,  a  black  band  will  appear.  It  is  the  point  of  compensation. 
A  quartz  wedge  must  be  carefully  cut  to  avoid  breaking  the  thin  edge, 
and  it  should  be  mounted  between  glasses  to  protect  it.  The  smaller  the 

angle  of  the  wedge,  the  broader  will  be 


J  (  I ,  j  the  interference  bands.     It  is  very  desir- 

^^2Md-^-3rd-^iltJ^        i     able   to   have   several   wedges;  one   with 
|  |     broad  bands  from  the  first  to  the  third 

order,    and    one    from    the  third  to  the 

FIG.  449. — Simple  quartz  or  gypsum  wedge. 

seventh.     The  orientation  of  the  vibration 

directions  in  the  wedges  should  be  the  same  as  that  in  the  mica  and  gypsum 
plates,  to  avoid  confusion. 

Like  the  mica  plate,  the  quartz  wedge  was  first  used  by  Biot.1  It  was 
later  made  use  of  by  de  Senarmont,2  but  subsequently  seems  to  have  been 
forgotten  until  Sorby3  announced  it  as  new  in  1877. 

296.  Fedorow  Mica  Comparator. — The  Fedorow  mica  comparator  is 
built  up  of  sixteen  rectangular  quarter  undulation  mica  plates,  each  2  mm. 
shorter  than  the  preceding.     It  is  described  in  full  below. 4     For  the  deter- 
mination of  the  optical  character  of  the  elongation  it  is  used  like  a  quartz 
wedge. 

297.  Wright  Combination  Wedge. — The  difficulty  with  ordinary  quartz 
wedges  is  that  it  is  impossible  to  grind  the  front  edge  sufficiently  thin  to  give 
a  dark  band  and  as  a  result,  upon  insertion,  the  color  rises  abruptly  to  about 
a  quarter  order.     To  overcome  this  objection,  Wright5  combined  a  quartz 
wedge  (b,  Fig.  450),  having  its  fast  ray  parallel  to  the  long  direction  of  the 

1  J.  B.  Biot:  Memoire  sur  les  proprietes  physiques  que  les  molecules  lumineuses  acquier- 
ent  en  traversant  les  cristaux  doues  de  la  double  refraction.  Lu  22  mai,  1814.  Mem.  Acad. 
France,  Annee  1812.  Paris,  1814.  31-38. 

Idem:  Traite  de  physique.     Paris,  1816,  IV,  420-422,  543-566. 

2H.  de  Senarmont:  Op.  cit.,  1851,  401. 

3  H.  C.  Sorby:    On  a  new  arrangement  for  distinguishing  the  axes  of  doubly  refracting 
substances.    Mon.  Microsc.  Jour.,  XVIII  (1877),  209-211. 

4  Art.  308. 

5  Fred.  Eugene  Wright:    Die  foyaitisch-theralitischen  Eruptivgesteine  der  Insel  CaboFrio, 
Rio  de  Janeiro,  Brasilien.     T.M.P.M.,  XX  (1901),  footnote,  p.  275. 

Idem:  A  new  combination  wedge  for  use  with  the  petro  graphical  microscope.  Jour.  Geol., 
X  (1902),  33-35- 


ART.  298]          VIBRATION  DIRECTIONS  IN  MINERAL  PLATES 


367 


wedge,  with  a  second  order  green  selenite  plate  (c)  in  which  the  fast  ray 
vibrates  at  right  angles  to  this  direction.  By  this  arrangement  compensation 
is  produced  at  about  the  center  of  the  wedge  where  a  dark  band  appears, 
to  the  right  and  left  of  which  the  interference  colors  rise.  Minerals  seen 
through  the  dark  bar  will  have  the  same  interference  colors  as  though  the 
wedge  were  not  there,  but  on  shoving  the  wedge  either  way  the  colors  gradu- 


FIG.  450. — Wright's  combination  wedge.     (Fuess.) 

ally  rise.  At  one  end  of  this  accessory,  for  convenience,  a  first  order  red  (a) 
is  added.  As  originally  described  this  gypsum  plate  was  separated  from  the 
wedge  by  an  open  space,  and  not  as  shown  in  the  figure. 

The  combination  wedge  was  later  made  of  a  quartz  wedge  and  a  quartz 
plate,  and  was  improved *  by  making  the  upper  and  lower  faces  parallel 
(Fig.  451)  thus  causing  no  displacement  of  the  image.  It  likewise  had  en- 
graved upon  the  upper  surface  a  scale  divided 
into  o.i  mm.,  the  wedge  being  so  calculated 
that  the  reading  gave  directly  the  difference 
in  fjifj.  in  the  retardation  of  the  wave.  In  order 
that  the  divisions  of  the  scale  may  be  seen, 
this  wedge  must  be  inserted  in  the  focal  plane 
of  the  ocular. 


FIG.    451. — Improved   combination 
wedge. 


a 

b 

c 

298.  Johannsen  Quartz-mica  Wedge. — The  interference  colors  of  the 
Wright  wedge  rise  abruptly  to  the  second  order,  no  matter  which  end  is 
inserted  first,  and  then  fall  to  zero.  In  the  wedge  described  by  Johannsen,2 
which  is  made  on  a  similar  principle,  this  does  not  occur,  and  the  transition 

from   the    interference    color   of   the 

/'  "x mineral  to  that  of  the  wedge  is  im- 
perceptible. The  wedge  (Fig.  452) 
consists  of  a  carriage,  exactly  fitting 
the  slot  above  the  objective,  and  per- 
manently retained  in  the  tube  of  the 
microscope  by  means  of  two  end  screws 

like  those  holding  the  Bertrand  lens  bar  in  the  Fuess  microscope.  At  one 
end  is  a  square  of  gypsum  (a)  giving  red  of  the  first  order;  b  is  an  opening, 
and  c  is  a  first  to  fourth  order  quartz  wedge  underlaid  by  a  mica  plate, 

1  Fred.  Eugene  Wright:    A  new  ocular  for  use  with  the  petrographic  microscope.     Amer. 
Jour.  Sci.,  XXIX  (1910),  416-417. 

2  Albert  Johannsen:    Some  simple  improiements  for  a  petro  graphical  microscope.     Amer. 
Jour.  Sci.,  XXIX  (1910),  436. 


FlG.  452. — Johannsen  quartz-mica  wedge. 


368  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  298 

the  two  minerals  having  their  c  directions  at  right  angles  to  each  other. 
The  thickness  of  the  mica  is  such  that  it  exactly  compensates  the  end  of 
the  wedge  at  the  edge  of  the  opening  b,  and  as  a  result,  when  the  wedge 
is  shoved  forward,  it  begins  at  total  darkness,  just  as  though  the  wedge 
were  infinitely  thin  at  this  end,  and  gradually  increases  to  the  fourth  order. 
A  spring  s,  attached  to  the  side  of  the  microscope  tube  or  within  it, 
presses  against  the  carriage  and  produces  enough  friction  to  hold  it  wherever 
placed.  When  the  opening  b  is  centered,  the  spring  drops  into  a  rounded 
notch  as  shown. 

For  the  determination  of  the  character  of  the  elongation  of  a  mineral, 
the  wedge  might  be  further  improved  by  making  it  in  two  parts,  like  the 
Evans  double  wedge,1  one  with  its  long  direction  parallel  to  the  c  axis,  and 
the  other  at  right  angles  to  it.  If  ground  to  the  same  slope,  both  would 
begin  at  absolute  darkness.  Since  the  slow  ray  would  be  vibrating  parallel 
to  the  long  direction  in  one,  and  at  right  angles  to  it  in  the  other,  when  placed 
above  a  mineral  section  in  the  position  of  maximum  illumination,  the  former 
would  compensate  with  minerals  having  negative  elongation  and  would 
show  a  dark  bar,  and  the  latter  would  compensate  with  those  having  positive 
elongation  and  show  the  bar.  If,  further,  a  scale  were  engraved  on  the 
upper  surface,  showing  directly  the  values  of  the  retardations,  it  would  be 
of  still  greater  value.  To  read  the  divisions  on  the  scale,  if  inserted  in  the 
usual  slot  above  the  objective,  it  would  only  be  necessary  to  insert  the 
Bertrand  lens.  If  inserted  in  the  focal  plane  of  the  ocular,  as  in  the  Seiden- 
topf 2  compensator,  the  divisions,  likewise,  could  be  directly  read. 

1  Art.  315. 

2  Art.  316. 


CHAPTER  XXV 
DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE 

299.  Birefringence. — It  has  been  shown1  that  the  retardation  in  a  doubly 
refracting  mineral  is  expressed  by  R  =  M(n%— HI),  and  that  the  interference 
color  produced,  when  white  light  is  used,  depends  upon  this  value;  n%  and 
«i,  in  the  equation,  being  the  refractive  indices  in  two  directions  at  right 
angles  to  each  other.     As  the  thickness  of  the  section  or  the  difference  between 
HZ  and  n\  increases,  so  does  the  color.     The  thickness  of  section  may  be  taken 
as  of  any  value.     If  it  is  considered  unity,  the  resulting  value  for  R,  that  is 
nz—nij  depends  only  upon  the  kind  of  mineral  and  upon  the  orientation  of 
the  section  writh  respect  to  the  principal  vibration  directions.     If  n^  and  n\ 
are  taken  as  the  values  of  the  indices  of  refraction  along  the  maximum  and 
minimum  ease  of  vibration  directions  in  any  mineral,  the  difference  between 
them  will  be  a  measure  of  the  maximum  double  refraction  or  birefringence  of 
that  mineral.     Just  as  is  the  refractive  index,  so  also  is  the  value  for  the  maxi- 
mum birefringence  characteristic  for  any  mineral.     It  is  expressed  in  positive 
uniaxial  crystals  by  e—  cu,  and  in  negative  by   o>— e.     In  biaxial  crystals 
it  is  expressed  by  7— or,  but  besides  this  maximum  value  there  are,  of  course, 
two  other  characteristic  values,  namely,  7— 0  and  0—  a. 

The  value  of  the  maximum  birefringence  of  a  mineral  may  be  determined 
by  computation  from  the  formula  just  given  if  the  refractive  indices  and  the 
thickness  of  the  section  are  known.  It  may  also  be  calculated  if  the  orienta- 
tion of  the  section  and  its  birefringence  or  indices  are  known.2 

In  practice,  the  double  refraction  of  a  mineral  is  usually  determined  by 
measuring  the  thickness  of  the  section  and  determining  the  point  of  compen- 
sation.3 The  simplest  accessory  for  this  purpose  is  the  compensating  wedge 
already  described. 

300.  Compensating  Wedge  for  the  Determination  of  Birefringence.— 

The  method  for  determining  the  birefringence  of  a  mineral  by  means  of  the 
quartz  or  gypsum  wedge  follows  directly  from  the  method  given  above  for 
determining  the  vibration  directions  of  a  crystal.  If  an  unknown  mineral 
is  placed  upon  the  stage  of  the  microscope,  between  crossed  nicols,  and  it  is 

1  Art,  280,  supra. 

2  Art.  288,  supra. 

3  For  the  effect  of  dispersion  on  double  refraction  see: 

C.    Hlawatsch:  Bestimmung  der   Doppelbrechung  fur  lerschiedene  Farben  an  einigen 
Mineralien.     T.  M.  P.  M.,  XXI  (1902),  107-156. 
24  369 


370  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  301 

turned  45°  off  extinction,  it  will  be  in  its  position  of  maximum  illumination. 
A  quartz  wedge  is  now  inserted  above  the  mineral,  and  the  change  of  colors 
noted.  If  they  ascend  in  the  scale,1  that  is,  pass  through  yellow  to  red 
to  violet  to  blue  to  green  to  yellow,  etc.,  the  vibration  directions  of  mineral 
and  wedge  are  clearly  parallel.  The  mineral  is,  therefore,  rotated  through 
90°  and  the  wedge  again  inserted.  The  order  of  the  change  of  colors  will 
now  be  reversed,  and  if  the  section  be  not  too  thick  or  the  birefringence  of 
the  mineral  too  high,  a  point  of  compensation  will  be  reached.  Beyond  the 
dark  bar  of  compensation  the  colors  will  again  appear,  but  in  ascending 
order.  When  compensation  occurs,  the  wedge  should  be  held  in  place  and 
the  mineral  removed  from  the  stage.  The  interference  color  shown  by  the 
wedge  should  now  be  the  same  as  that  which  originally  was  shown  by  the  min- 
eral, except  so  far  as  the  latter  may  have  been  made  abnormal  by  color,  dis- 
persion, etc.  As  the  wedge  is  slowly  withdrawn,  the  sequence  of  colors  may 
be  noted  and,  by  counting  the  number  of  times  a  color  recurs  and  making 
comparsion  with  a  color  chart,2  its  exact  position  in  the  scale  may  be  deter- 
mined. The  value  obtained,  however,  is  that  of  the  retardation,  and  not 
the  true  value  of  the  birefringence,  for  this  is  influenced  by  the  thickness  of 
the  section,  as  may  be  seen  from  the  equation,  R  =  M  (n2  —  ni).  To  deter- 
mine the  thickness  M,  recourse  may  be  had  to  the  method  of  the  Due  de 
Chaulnes  or  to  any  of  the  other  methods  suggested  in  Art.  208.  In  de 
Chaulnes  method,  however,  it  is  necessary  to  know  the  refractive  index  of 
the  mineral.  This  may  be  unknown  in  the  mineral  under  observation  for 
birefringence,  but  in  a  rock  section,  adjacent  to  the  unknown  mineral,  there 
probably  is  some  mineral  which  is  known.  In  the  known  mineral,  then,  the 
determination  of  thickness  may  be  made.  By  dividing  the  value  of  the  re- 
tardation, as  obtained  with  the  quartz  wedge,  by  the  thickness,  a  retarda- 
tion value  for  unity  may  be  obtained,  and  from  this,  by  comparison  with  a 
table,  the  value  of  the  birefringence  of  the  mineral. 

301.  Michel-Levy  Chart  of  Birefringences  (1888).— The  best  table  for 
the  comparison  of  interference  colors  is  that  devised  by  Michel-Levy 3  and 
shown  in  outline  in  Fig.  453.  In  the  original  chart  there  are  shown,  from 
left  to  right,  colors  as  nearly  as  possible  like  those  produced  by  increased 
retardation,  the  values  of  which  [M(nz  —  n\)]  are  shown  in  millionths  of 

1  Cf.  Newton's  scale,  Arts.  276-277. 

2  Art.  301.     The  colored  plate  of  birefringences,  originally  given  by  Levy  and  Lacroix, 
has  been  reproduced  frequently  and  may  be  found  in  Rosenbusch-Wiilfmg,  Duparc  and 
Pearce,  Iddings,  or  Johannsen.     Cf.  Fig.  451. 

3  A.  Michel-Levy  et  Alf.  Lacroix:  Les  mineraux  des  roches.     Paris,  1888,  plate  i. 

See  also  A.  Michel-Levy:  Mesure  du  pouvoir  birefringent  des  mineraux  en  plaque  mince. 
Bull.  soc.  min.  France,  VI  (1883),  143-161. 

Idem:  Note  sur  la  birefringence  de  quelques  mineraux;  application  a  V etude  des  roches  en 
plaques  minces.  Ibidem,  VII  (1884),  43-47. 


ART.  301    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE 


371 


Thickness  in  Millimeters 

i 


III 


IV  1 


%  Light  Carmine 
Light  Purple 
Grayish  Violet 


2,000 


1 I  H        !\l    IM I    l\    \l 

\   \      \ 


I  2  t  £ 


FIG.  453. — Outline  of  Michel-LeVy's  chart  of  birefringences,  the  positions  of  the  colors  modified  accord- 
ing to  the  Kraft  scale  for  a  clear  sky. 


372  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  301 

millimeters  by  the  abscissae.  The  ordinates  represent  thickness  of  section 
(M).  The  value  of  unit  birefringence,  n^—ni  (that  is,  7  —  a  or  co  —  e), 
remains  constant  for  any  mineral,  but  as  the  section  increases  in  thickness  so 
does  the  retardation  increase,  as  may  be  seen  from  the  retardation  equation. 
The  diagonal  lines  in  the  diagram  represent,  therefore,  the  retardations  pro- 
duced by  sections  of  different  thicknesses. 

From  this  chart  one  may  determine  not  only  the  order  of  birefringence 
of  a  mineral,  but  the  thickness  of  a  section  as  well,  provided  some  mineral 
contained  in  the  slide  is  known.  For  example,  in  a  granite  the  fragments 
of  quartz  are  easily  recognized.  Among  these,  note  the  highest  inter- 
ference color.  If  the  slide  contains  numerous  fragments  it  is  probable 
that  this  is  a  section  parallel  to  the  optic  axis,  and  its  birefringence  has  the 
maximum  value,  0.009.  ^n  the  chart  this  value  is  given  by  a  diagonal  line. 
Follow  it  down  toward  the  lower  left  corner  until  it  intersects  the  vertical 
line  giving  the  interference  color  shown  in  the  slide.  The  ordinate  at  the 
point  of  intersection  represents  the  thickness  of  the  slide,  and  its  value  may 
be  found  by  following  out  the  horizontal  line,  through  the  intersection, 
to  the  scale  at  the  left.  The  value  there  found  is  the  thickness  of  the  slide 
in  millimeters. 

To  determine  the  birefringence  of  an  unknown  mineral  the  method  is 
as  follows:  Determine  the  thickness  of  the  section  by  any  of  the  methods 
given  in  Art.  208,  or  by  means  of  the  birefringence  of  some  known  mineral 
by  the  method  just  indicated.  It  is  advisable  to  make  use  of  a  known  mineral 
fragment  lying  as  near  as  possible  to  the  unknown,  since  there  may  be  a 
slight  difference  in  the  thickness  of  different  parts  of  the  slide.  No  hesitation 
should  be  felt,  however,  in  using  this  method  for  fear  that  the  section  may 
be  unequally  ground,  for  differences  in  thickness  can  be  recognized  readily 
by  the  variation  in  the  interference  colors  in  different  parts.  Having 
determined  the  thickness  of  the  slide,  determine  the  highest  interference 
color  in  any  fragment  of  the  unknown  mineral.  Take  the  intersection  of 
the  horizontal  line  of  thickness  in  the  chart  with  this  color.  The  diagonal 
line  passing  through  this  point  of  intersection  indicates  the  birefringence  of 
the  unknown  mineral. 

For  example,  in  a  slide  of  a  basalt  there  are  many  fragments  of  labradorite 
whose  maximum  birefringence  is  0.008.  If  its  highest  interference  color  in 
the  rock  slice  is  pale  straw  color,  the  thickness  of  the  slide  is  0.034.  An 
unknown  mineral  in  the  same  rock  slice  has  an  interference  color  of  blue  of 
the  third  order.  The  diagonal  line  crossing  the  point  of  intersection  is 
0.035,  which  is  the  value  of  the  maximum  birefringence  of  meionite,  humite, 
and  olivine.  From  other  characteristics  of  the  minerals  we  can  easily 
separate  these  three  and  determine  the  unknown  mineral  as  olivine. 


ART.  302]    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE 


373 


TABLE 

OF  MAXIMUM 

BIREFRINGENCES 

Rutile  

0.287 

Hedenbergite  

0.019 

Micaceous  hematite  

0.28 

Lawsonite  

0.019 

Siderite  

0.238 

Glaucophane  

0.018 

Magnesite  

O.2O2 

Monticellite  

0.017 

Dolomite  

O.I7Q 

Spodumene  

0.016 

Calcite  

o.  172 

Common  hornblende  

0.016 

Brookite  

0.158 

Mizzonite  

0.015 

Aragonite  

0.156 

Wollastonite  

0.015 

Titanite  

0.145 

Anorthite  

'  0.013 

Cassiterite  

0.096 

Serpentine  

0.013 

Anatase  

0.073 

Dipyr  

0.013 

Basaltic  hornblende  

0.072 

Hypersthene  

0.013 

Zircon  

0.062 

Cornerupine  

0.013 

Grunerite  

0.056 

Natrolite  

0.012 

Astrophyllite  

0.055 

Disthene  

O.OI2 

Favalite  

0.050 

Johnstrupite  

O.OI2 

.^Egirite  

0.050 

Mosandrite  

O.OI2 

Talc  
Diaspore  

O.O5O 
0.048 

Hydronephelite  
Laumontite  

O.OI2 
O.OI2 

Monazite  

o  .  045 

Andalusite  

O.OII 

Anhydrite  

o  .  044 

Antigorite  

O.OII 

Datolite  

o  .  044 

Clinochlore  

O.OII 

Phlogopite  

o  .  044 

Dumortierite  

O.OII 

Biotite  

o  .  040 

Gypsum  

O.OIO 

Lavenite  

o  .  040 

Axinite  

O.OIO 

Muscovite  

0.038 

Staurolite  

O.OIO 

Pectolite  

0.038 

Ottrelite  

O.OIO 

Lazulite  

0.036 

Epistilbite  

O.OIO 

Olivine  

0.035 

Albite  

O.OIO 

Humite  

0.035 

Quartz  .  .           .             

o  .  009 

Meionite  

0.035 

Corundum  

o  .  009 

Prehnite  

0.033 

Enstatite  

o  .  009 

Titanolivine  

0.033 

Bronzite  

o  .  009 

Pistacite  

0.032 

Cordierite  

o  .  009 

Chondrodite  

0.032 

Topaz  

o  .  009 

Orthite  

0.032 

Zoisite  

o  .  009 

Diopside  

0.029 

Labradorite  

o  .  008 

Jadeite  

,  0.029 

Kaolin  

o  .  008 

iEgirite-augite  

0.029 

Clinozoisite  

o  .  008 

Cancrinite  

0.028 

Scolecite  

o  .  007 

Thomsonite  

0.028 

Heulandite  

0.007 

Actinolite  

0.027 

Orthoclase  

o  .  006 

Tremolite  

0.026 

Gehlenite  

o  .  006 

Wohlerite  

0.026 

iEnigmatite  

0.006 

Rosenbuschite  

0.026 

Stilbite  

o  .  006 

Tourmaline  

0.025 

Sapphirine  

o  .  005 

Augite  

0.025 

Melilite  

0.005 

Anthophyllite  

0.024 

Nephelite  

o  .  005 

Hydrargillite  

0.023 

Riebeckite  

o  .  004 

Carpholite  

O.O22 

Apatite  

o  .  004 

Sillimanite  

O.O22 

Eucolite  

o  .  003 

Brucite  

O.02I 

Phillipsite  

o  .  003 

Gedrite  

O.O2I 

Eudialite  

O.OO2 

Barkevikite  

O.O2I 

Tridymite  

0.002 

Alunite  

O.O2O 

Vesuvianite  

O.OO2 

Melinophane  
Pargasite  

C.020 
O.02O 

Pennine  
Leucite  

O.O02 
O.OOI 

302.  Babinet  Compensator. — One  of  the  most  delicate  instruments  for 
determining  the  birefringence  of  a  mineral  is  the  Babinet  compensator. 


374 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  302 


Unfortunately  the  methods  for  determining  the  thickness  of  a  section  are 
not  so  delicate  as  that  for  determining  the  birefringence  by  this  instrument, 
and  since  this  is  a  factor  in  obtaining  the  result,  the  advantage  is  somewhat 
lessened. 

The  Babinet1  compensator  consists  essentially  of  a  Ramsden  ocular, 
beneath  which  are  arranged  two  right 
angled  quartz  wedges  with  equal  slopes 
and  with  their  inclined  faces  toward 
each  other  (Fig.  454).  One  of  these 
wedges  is  movable  by  means  of  a  mi- 
crometer screw,  and  one  is  stationary. 
The  vibration  directions  of  the  two 
wedges  lie  at  right  angles  to  each  other, 


FIG.  454. — Section  through  the  wedges  of  a 
Babinet   compensator. 


FIG.  455. — Babinet   compensator.     2/3 
natural     size.     (Fuess.) 


one  being  cut  with  the  long  direction  parallel  to  the  optic  axis,  and  one 
at  right  angles  to  it.  In  the  Fuess  instrument  (Fig.  455),  the  lower 
wedge,  which  has  a  length  of  25  mm.,  is  the  one  which  is  movable.  Upon 
the  stationary  wedge  there  is  engraved  a  cross,  whose  arms  make  an  angle 
of  30°  with  each  other,  and  whose  center  is  on  the  axis  of  the  microscope. 

1  M.  J.  Jamin:  Memoire  sur  la  reflexion  a  la  surface  des  corps  transparents.  Ann.  d. 
chim.  etphys.,  XXIX  (1850),  263-304,  especially  271-274. 

A.  Bravais:  Beschreibung  ernes  neuen  Polariskops  und  Unlersuchung  iiber  die  schwachen 
Doppelbrechungen.  Pogg.  Ann.,  XCVI  (1855),  395-414,  especially  pages  404-409. 

Idem:  D'un  nowoeau  polariscope  et  recherches  sur  les  doubles  refractions  peu  energiques. 
Ann.  d.  chim.  et  phys.,  XLIII  (1855),  129-149. 

G.  Quincke:  Optische  Experimental-Untersuchungen.  II.  Ueber  die  elliptische Polari- 
sation des  bei  totaler  Reflexion  eingedrungenen  oder  Zuriickge-worfenen  Lichtes.  Pogg.  Ann., 
CXXVII  (1866),  203-212. 

J.  Mace  de  Lepinay:  Recherches  experimentales  sur  la  double  refraction  accidentelle. 
Ann.  d.  chim.  et  phys.,  XIX  (1880),  5-90. 

Karl  E.  Franz  Schmidt:  Zur  Theorie  des  Babinet' schen  Compensators.  Wiedem.  Ann., 
XXXV  (1888),  360-369. 

Idem:  Zur  Konstruktion  des  Babinet' 'schen  Kompensators.  Zeitschr.  f.  Instrum.,  XI 
(1891),  439-444- 

J.  Mace  de  Lepinay:  Sur  la  localisation  des  f  ranges  des  lames  cristallines .  Ann.  d.  Phys., 
X  (1891),  204-213.* 

C.  Leiss:  Die  optischen  Instrumente,  etc.,  Leipzig.  1899,  223. 

Thomas  Preston:  The  theory  of  light.    London,  3d  ed.,  1901,  410-415. 

F.  Becke:  Denkschr.  Akad.  Wiss.  Wien,  LXXV  (1904),  58-* 

Rosenbusch-Wulfing:  Mikroskopische Physiographic,  Stuttgart,  4  Aufl.,  1904,  Ii}  284-289. 

Duparc  et  Pearce:  Traite  de  technique  min.  et  petrog.    Leipzig,  1907,  206-211. 

A.  E.  H.  Tutton:  Crystallography.    London,  1911,  859-862. 


ART.  302]    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE  375 

This  cross  serves  for  a  starting  point  from  which  to  measure  the  displacement, 
which  may  be  read  to  0.005  mm-  by  means  of  the  vernier. 

The  Babinet  compensator  is  inserted  in  the  tube  of  the  microscope  instead 
of  an  ocular,  and  is  so  placed  that  the  axes  of  the  quartz  wedges  make  angles 
of  45°  with  the  vibration  plane  of  the  polarizer.  The  analyzer  in  the  tube 
of  the  microscope  is  not  inserted,  but  a  cap  nicol  is  placed  above  the  eyepiece. 
In  such  a  position,  and  with  the  vernier  set  at  zero,  a  black  bar  appears  in 
the  center  of  the  field  and,  on  either  side  of  it,  a  series  of  colored  bars  in 
white  light,  or  black  bars  separated  by  white  spaces  in  monochromatic 
light.  These  bars  are  caused  by  the  separation  into  two  rays  of  the  polarized 
light  which  enters  from  below  at  right  angles  to  the  wedge.  One  of  these  rays 
has  vibrations  parallel  to  the  vibration  direction  of  the  lower  wedge,  the 
other  vibrates  at  right  angles  to  it.  Upon  passing  into  the  upper  wedge,  the 
vibration  directions  remain  the  same  but  the  velocities  are  different,  the 
slow  ray  of  the  first  becoming  the  fast  ray  of  the  second,  and  vice  versa. 
At  the  center  of  the  wedge,  where  the  thickness  of  each  is  the  same,  the 
sum  of  the  two  vibrations  in  opposite  directions  will  be  zero,  for  R  =  M(nz  — 
HI)  —M(n2—ni)  =  o.  If  the  lower  wedge  is  moved  so  that  its  thickness  is  MI 
the  equation  becomes  Ri  =  (Mi— M)  (n2—ni). 

Since  the  scale  and  vernier  are  graduated  to  millimeters,  it  is  necessary 
to  determine  the  relation  between  the  retardation  and  the  lateral  displace- 
ment. This  may  be  accomplished  very  simply  by  setting  the  cross-hairs 
on  the  dark  band  by  white  light  and  then,  by  monochromatic  light,  measur- 
ing the  distance  through  which  the  wedge  must  be  moved  to  cause  the 
cross-hairs  to  coincide  with  one  of  the  adjacent  dark  bars.  This  displace- 
ment represents  iX  for  whatever  light  was  used.  If  this  was  sodium  light, 
then  the  number  of  divisions  through  which  the  drum  D  was  turned  cor- 
responds to  589^1/1,  and  each  division  to  -~-  =K,  the  constant  for  the  instru- 
ment with  sodium  light.  Measurements  should  be  made  for  the  value  of  K 
between  all  the  dark  bars,  and  if  the  instrument  is  properly  made,  these  values 
should  be  the  same.  If  they  are  not,  a  curve  may  be  drawn  to  represent 
the  value  of  retardation  for  one  division  of  the  drum  at  every  point  of  the 
wedge. 

To  determine  the  birefringence  of  any  mineral  with  the  Babinet  com- 
pensator, the  instrument  should  be  set  up  as  described  above,  and  the  black 
bar  be  made  to  correspond  with  the  cross.  The  vernier  should  read  zero  at 
this  point.  If  the  graduations  are  in  miLimeters  it  is  of  little  importance 
whether  this  reading  is  correct  or  not,  for  the  displacement  may  be  de- 
termined by  the  difference.  If,  however,  the  graduations  are  in  up,  the 
zero  value  should  correspond.  When  a  mineral  section  is  placed  upon  the 
stage  and  it  is  turned  45°  off  extinction,  the  dark  bar  will  be  displaced  to  the 
right  or  left,  depending  upon  whether  the  vibration  directions  are  parallel 


376  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  303 

or  at  right  angles  to  those  of  the  comparator.1  To  bring  the  black  bar  back 
to  the  cross,  a  number  of  turns  of  the  drum  are  necessary.  From  the  calibra- 
tion previously  made,  the  amount  of  this  displacement  in  juju  may  be  de- 
termined. The  value  thus  obtained  represents  the  retardation  of  the  light 
in  passing  through  the  mineral  plate.  The  actual  value  of  the  birefringence 
may  be  determined  from  this  retardation  and  the  known  thickness  of  the 
slice  in  the  manner  described  for  the  quartz  wedge.2 

303.  Von  Chrustschoff  Twin  Compensator  (1896). — The  von  Chrust- 
schoff3  compensator  is  a  modification  of  that  of  Babinet.  Instead  of  a 
simple  wedge,  it  is  made  up  of  two  pairs  (Fig.  456),  the  orientation  of  the  vi- 
bration directions  being  such  as  would  occur  if  a  simple  Babinet  pair  were  cut 

longitudinally  and  one  pair  rotated 
1 80°  in  altitude  so  that  the  bottom 
becomes  the  top.  In  this  position 
the  two  parts  of  the  upper  pair  and 
the  two  parts  of  the  lower  pair  are 
FIG.  456.— Von  Chrustschoff  twin  compensator,  cemented  together,  making  artificial 

twin  wedges.     When  set  at  zero,  the 

black  bar  is  continuous  across  the  two.  If,  however,  a  mineral  is  placed 
beneath  it,  one-half  of  the  wedge  will  reduce  the  retardation  while  the 
other  half  will  add  to  it.  The  black  bar  will  thus  be  separated  into  two 
bars  equally  distant  from  the  center,  one-half  the  distance  between  them 
representing  the  retardation  A  scale  engraved  on  the  upper  surface, 
along  the  twinning  line,  permits  a  coarse  direct  reading  to  o.  i  mm.  displace- 
ment. The  fine  adjustment  is  by  means  of  a  screw,  which  may  be  calibrated 
in  a  manner  similar  to  the  Babinet.  The  results  obtained  with  this  instru- 
ment are  said  to  be  accurate  to  the  fourth  decimal  place. 

As  constructed  by  Fuess,4  the  two  upper  wedges  are  short  and  stationary, 
the  lower  movable  in  separate  mountings  and  so  arranged  that  with  one 
movement  of  the  screw  the  two  slide  with  equal  displacements  in  opposite 
directions.  Two  scales  are  provided;  one  with  divisions  of  o.i  mm.  is 
engraved  directly  above  the  wedge  and  appears  in  the  field  of  the  microscope, 
another,  for  accurate  measurements,  is  given  by  the  screw  micrometer  which 
reads  to  o.ooi  mm.  The  actual  amount  of  the  displacement  is,  of  course, 
double  that  given  by  the  micrometer,  since  each  wedge  has  traveled  the 
distance  recorded. 

1  For  the  effect  of  dispersion  produced  by  the  Babinet  comparator  on  the  dark  bar,  see 
C.  Hlawatsch:  Bestimmung  der  Doppelbrechung  fur  lerschiedene  Farben  an  einigen  Miner- 
alien.     T.  M.  P.  M.,  XXI  (1902),  107-156. 

2  Art.  300,  supra. 

3K.  von  Chrustschoff:    Abh.    d.    kais.    russ.  min.   Gesell.,  Ser.  II,  XXXIV  (1896), 
165-169. *     Review  Ueber  einen  Zwillingscompensator .  Zeitschr.  f.  Kryst,  XXX  (1899),  389. 
4  C.  Leiss:  Die  optischen  instrumente,  etc.,  Leipzig,  1899,  223-224. 


ART.  304]    DETERMINATION  OF  THE  ORDER  OF  BIREFRIXGEXCl. 


377 


304.  Michel-Levy  Comparator  (1883).— The  Michel-Levy1  comparator 
(Fig.  457)  is  another  device  for  determining  the  double  refraction  of  minerals. 
It  differs  in  principle  from  the  preceding  in  that  the  determination  is  made 
by  comparison  and  not  by  compensation  and  is,  perhaps,  quite  as  good  for 
very  small  fragments  of  colorless  minerals. 

The  internal  arrangement  of  the  instrument  is  shown  in  the  cross-section 
Fig.  458.  E-F  is  an  ocular,  in  the  place  of  whose  diaphragm  there  is  inserted 
a  prism  P,  silvered  on  the 
slanting  face  with  the  excep- 
tion of  a  small  circular  open- 
ing in  the  center.  Over 
this  clear  space,  which  is  on 
the  axis  of  the  microscope, 
a  second  prism  P' ',  cut  from 
a  cylinder  of  glass,  is  cemen- 
ted by  Canada  balsam. 
There  thus  passes  to  the  eye 
along  the  axis  of  the  micro- 
scope a  beam  of  light  coming 

from    the    rOCk    Section.      At  Fic.  457._Michel-L6vy  comparator.     (Xachet.) 

the  same  time  the  periphery 

of  the  field  is  illuminated  by  the  light 

which  is  reflected  from  the  mirror  M 

and  passes    through    the   prism   C,    a 

quartz  wedge  A,  a  diaphragm  D,  and 

the  two  nicol  prisms  N'  and  N  whose 

vibration  planes  lie  at  45°  to  those  of 

the  wedge.    The  latter  may  be  moved 

across  the  field  of   view  by  means  of 

the  screw  shown  in  Fig.  457,  and  the 

amount   of  the  movement   read   from 

the  scale  and  vernier.    Between  the  second  nicol  N  and  the  prism  P,  the 

lens    B    converts    the   light   into  parallel  rays  which  are  reflected   from 

the  silvered  back  of  the  prism  P  to  the  eye.    If  the  two  nicols,  N  and  N', 

are  crossed,  the  periphery  of  the  field  of  view  of  the  microscope  will  show 

interference  colors  produced  by  the  quartz  wedge,  increasing  in  the  scale  as 

the  wedge  is  moved  forward.    On  looking  through  the  microscope,  then, 

the  center  of  the  field  will  be  colored  by  the  mineral,  which  should  be  turned 

1  A.  Michel-Levy:    Mesure  du  pouvoir  birefringent  des  mineraux  en  plaque  mince.     Bull, 
soc.  min.  France,  VI  (1883),  143-161. 

Levy  et  Lacroix:  Les  mineraux  des  roches.     Paris,  1888,  54-59. 

R.  Fuess:   Quarzkeilcomparator  nach  Michel-Levy.     Neues  Jahrb.    B.B.,    VII    (1889), 
77-79- 

C.  Leiss:  Die  optischen  Instrumente,  etc.     Leipzig,  1899,  224. 


FIG.  458. — Section  through    the    Michel-Levy 
comparator. 


378  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  305 

45°  off  extinction,  and  the  border  by  the  quartz  wedge  of  the  comparator. 
To  determine  the  color  of  the  former  all  that  is  necessary  is  to  move  the 
wedge  until  the  boundaries  disappear  and  the  colors  are  the  same.  If  the 
interference  color  of  the  mineral  is  less  than  the  first  order  yellow,  it  is 
advisable  to  increase  it  by  placing  a  quarter  undulation  or  unit  retardation 
plate  above  it,  in  parallel  position,  allowing  for  this  increase  in  computing 
the  result. 

The  instrument,  as  made  by  Fuess,  differs  somewhat  from  that  described 
above  in  that  both  nicols  may  be  rotated  and  the  quartz  wedge  thrown  out 
of  the  field.  The  latter  improvement  is  important  in  equalizing,  prior  to 
an  observation,  the  illumination  through  the  microscope  and  through  the 
comparator.  When  the  wedge  is  removed,  the  periphery  of  the  field  will 
appear  dark.  By  closing  the  iris  diaphragm  below  the  stage,  more  or  less, 
the  amount  of  light  in  the  center  of  the  field  may  be  reduced  to  the  same 
amount  as  that  reflected  through  the  wedge,  so  that  the  whole  field  will 
appear  equally  dark. 

The  calibration  of  the  instrument  is  performed  in  a  manner  similar  to 
the  calibration  of  the  Babinet  compensator.  Two  readings  are  taken,  one 
for  the  violet  of  the  first  order  with  a  retardation  of  575  w,  and  one  for  the 
second  order  with  a  retardation  of  1 128  ///*.  If  t  and  t'  represent  the  readings 
on  the  scale,  one  division  d  will  be  represented  by  the  equation 

1128-575 
d-     -j^-w 

PROBLEMS 

Calibrate  the  Michel-Levy  comparator. 

Determine  the  interference  color  shown  by  a  section  of  quartz  cut  parallel  to 
the  optic  axis. 

Determine  the  actual  birefringence  of  an  unknown  mineral  in  a  rock  section 
containing  a  known  mineral. 

305.  Fedorow  Method  for  Determining  Low  Interference  Colors  (1892). 
• — We  saw1  that  the  interference  color  of  a  mineral  plate  was  different 
between  parallel  and  between  crossed  nicols.  Fedorow2  made  use  of  this 
difference  in  determining  the  value  of  interference  colors  of  the  lower  first 
order.  These  colors,  low  gray  and  white,  are  recognizable  with  difficulty 
between  crossed  nicols  but  are  readily  distinguishable  when  the  nicols  are 
parallel,  as  may  be  seen  from  the  table  given  in  Arts.  308,  276,  or  277. 

1  Arts.    276-277.     See  also  Art.  286. 

2  E.  von  Fedorow:    Mikroskopische  Beobachtung  bei  paralleler  Lage  der  Nicols.     Neues 
Jahrb.,  1892  (II),  69-70. 

Idem:  Zur  Bestimmung  der  Feldspathe  und  des  Quarzes  in  Dunnschli/en.  Zeitschr.  f. 
Kryst,  XXIV  (1894-5),  131. 


ART.  308]    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE  379 

306.  Cesaro    Wedge    (1893). — Cesaro's1   wedge   does  not  differ  in  its 
essentials  from  the  earlier  wedges,  but  the  determinations  are  made  by  com- 
pensation, not  to  darkness,  but  to  the  first  order  violet.     This,  according  to 
Cesaro,  is  most  sensitive  between  parallel  nicols  when  it  has  a  retardation  of 
281  fjifjL.     A  scale,  indicating  the  displacement,  is  attached  outside  the  ocular, 
and,  being  divided  into  millimeters,  must  be  calibrated.     The  drum  vernier 
reads  to  twentieths  of  millimeters. 

307.  Amann  Birefracto meter   (1895). — Another   instrument,  based  on 
compensation,    is  that  of  Amann.2    This   consists  essentially  of  a  quartz 
wedge  inserted  in  the  focal  plane  of  aHuygens' 

ocular  and  movable  by  means  of  a  screw.  The 
upper  surface  of  the  scale  is  divided  into  milli- 
meters, and  the  subdivisions  are  read  by  means 
of  the  vernier  drum.  As  constructed  by  Fuess 
(Fig.  459),  the  quartz  wedge  K  covers  but  half 
the  field  of  view.  It  is  attached  to  the  slides 
ss,  and  moved  by  the  drum  k.  The  glass  plate 
o,  to  the  lower  side  of  which  the  wedge  is  fas- 
tened, is  engraved  with  a  scale  divided  to  0.2 

mm.       The    long    Side    Of     the    Wedge     and     tWO     FIG.  459. — Amann  birefractometer. 
I*    i    ,.  j  ,,.         1  2/3  natural  size.      (Fuess.) 

parallel  lines  engraved  upon  a  thin  glass  serve 

as  cross-hairs.     /  is  a  lever  to  control  an  iris  diaphragm. 

To  determine  the  birefringence  of  a  mineral,  the  ocular  is  inserted  in  the 
tube  of  the  microscope  in  the  45°  position  and  a  cap  nicol  is  placed  over  it, 
resting  on  the  shoulder  T.  The  mineral  to  be  determined  is  rotated  to  45° 
off  extinction  and  with  its  fast  ray  at  right  angles  to  that  of  the  wedge.  The 
latter  is  screwed  forward  to  compensation  and,  from  previous  calibration, 
the  double  refraction  is  determined.  For  mineral  fragments  or  crystals 
around  the  periphery  of  a  rock  slice,  this  refractometer  may  be  used  as  a 
comparator,  since  the  wedge  covers  but  half  the  field.  With  ordinary  light 
it  may  be  used  as  a  screw  micrometer  ocular  for  measuring  distances. 

308.  Von  Fedorow  Comparator  (1895). — The  von  Fedorow3  comparator 

1  G.  Cesaro:  Sur  une  methode  simple  pour  mesurer  le  retard  des  mineraux  en  lames 
minces.  Bull.  Accad.  Roy.  Belgique,  Cl.  Sci.,  XXVI  (1893),  208-227. 

2J.  Amann:  Le  birefractometre  ou  oculaire-comparateur .  Zeitschr.  f.  wiss.  Mikrosk., 
XI  (1894),  440-454. 

C.  Leiss:  Compensator-Ocular  nach  J.  Amann.  Neues  Jahrb.,  B.B.,  X  (1895-6),  425- 
428. 

C.  Leiss:  Die  optischen  Instrumente,  etc.,  Leipzig,  1899,  226. 

3  E.  von  Fedorow:     Ueber  einen  Glimmer  comparator.     Zeitschr.  f.  Kryst.,  XXV  (1895), 

349-351- 

Idem:  Calibrirung  der  Glimmercomparatoren.     Ibidem,  XXVI  (1896),  251-254. 

Idem:  Die  Feldspalhe  des  Bogoslowsk'schen  Bergreriers.  Ibidem,  XXIX  (1897-98), 
611-613. 

C.  Leiss:  Die  optischen  Instrumente,  etc.    Leipzig,  1899,  211. 


380 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  80S 


consists  of  sixteen  rectangular,  quarter  undulation  mica  plates,  each  2  mm. 
shorter  than  the  preceding,  and  cemented  with  Canada  balsam  into  a  step- 
like  "wedge"  (Fig.  460).  There  is  thus  formed  a  compensator  ranging  in 
values  from  1/4  wave  length  to  four  wave  lengths  retardation,  each  step 
of  1/4  X  being  called  by  von  Fedorow  a  Levy  (written  L).  The  comparator 
is  used  in  a  manner  similar  to  a  quartz  wedge.  After  compensating,  the 
mineral  is  removed  and  the  number  of  the  step  determined.  If  the  Bertrand 

lens  is    inserted    and  the   nicols  left 
crossed,  the  separating  lines  between 

FIG.  46o.-Von  Fedorow  mica  compensator.        the  plates  may  be  distinctly  SCCn.     The 
Vertical  scale  greatly  exaggerated. 

wedge   may  be   purchased   with    the 

value  of  the  retardation  of  each  step  engraved  upon  the  cover-glass,  thus 
avoiding  the  necessity  of  counting  the  steps. 

The  retardation  and  interference  color  of  each  step  between  crossed  and 
parallel  nicols  are  as  follows: 


Order 

Crossed  nicols 

Retardation 

Order 

Parallel  nicols 

i 

Gray 

i 

i 

Pure  white  

2*C< 

i 

Dark  violet-brown 

i 
i 

Orange-yellow  
Orange-red 

382 

C  1  1 

;! 

Sky-blue 
Light  yellowish 

2 

Blue.  .  . 

6^7 

ji 

Canary-yellow 

2 

Green  .  . 

?6q 

i  ^ 

Yellowish  orange 

2 

Yellow.  . 

802 

2^ 

Intense  violet-blue 

2 

Orange-red  

IO2O 

2  ir 

Leek-green 

•2 

Indigo 

1  147 

2^' 

Chrome-yellow 

3 

3 

•2 

Smaragdite-green  
Lemon-yellow  
Orange  .  . 

1275 
I4O2 
I  ^3O 

2; 

3i? 

3^ 

Light  orange 
Pure  violet 
Pure  green 

4 
4 
4 

Violet-red  
Grass-green  
Greenish  yellow  

1657 
1785 
IQI2 

3^ 
3? 
4^ 

Light  yellowish 
Light  orange 
Light  reddish  blue. 

4 

Rose. 

2Q4O 

Light  greenish 

By  means  of  this  comparator  it  is  possible  not  only  to  determine  the 
birefringence  of  a  mineral  to  one  Levy  (1/4  X),  that  is,  to  the  retardation 
of  a  single  step,  but  even  to  1/8  Levy  (1/32  X). 

For  example,  if  one  reduces  a  mineral  to  darkness  by  means  of  this  com- 
pensator and  finds  one  step  dark  and  the  step  above  and  the  step  below 
equally  illuminated,  then  the  difference  of  phase  is  exactly  the  value  of  the 

step  N  -.    If  two  adjacent  steps  are  equally  bright,  the  value  is  intermediate 

N\ 

between  the  two,  and  a  half  step  must  be  added, hi/8X  (or  N'L-\-i/2L). 

4 
If  two  adjacent  steps  are  not  quite  equally  dark,  one  can  estimate  the  value 


ART.  308]    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE 


381 


by  comparison  with  the  two  beyond  the  dark  pair,  as  i  to  3,  or  1/4  of  one 
step  (1/4  L)  difference. 

A  method  of  increasing  the  delicacy  of  the  determinations  was  given  by  von 
Fedorow1  in  1898.  He  prepared,  first,  1/4  Levy  (1/16  X)  retardation  plates  by 
splitting  mica  into  very  thin  lamellae,  and  from  those  having  like  interference  colors, 
he  selected  four  that  just  compensated  a  quarter  undulation  mica  plate  when 
superimposed  in  parallel  positions  or  sixteen  that  compensated  a  first  order  red. 
From  such  1/4  L  micas,  two  small  rectangles  were  cut  as  shown  in  Fig.  461,  in  which 
the  dotted  lines  represent  the  steps  of  a  von  Fedorow  compensator  and  the  heavy 
lines  the  mica  accessory  plate.  The  width  of  each  plate  is  one-half  that  of  the  com- 
parator, and  the  length  4  mm.,  so  that  each  covers  two  steps.  The  two  micas  are 
cemented  between  cover-glasses,  one  with  its  vibration  directions  parallel,  the 


III 


FIG.  462. 


FIG.  463. 


FIG.  461. 

FIG.  461. — Von  Fedorow  mica  accessory  plate. 

FIG.  462. — A  mineral  differs  by  i  L  from  a  step  of  the  compensator. 
FIG.  463. — A  2  1/2  L  mineral  differs  by  1/2  L  from  one  step  of  the  compensator. 

other  at  right  angles,  to  that  of  the  comparator.  By  this  means  it  is  possible  to 
make  readings  to  1/8  L  (1/32  X)  with  ease  and  accuracy. 

In  using  this  comparator  it  is  placed  above  a  mineral  in  opposite  phase,  and  the 
latter  is  reduced,  by  compensation,  as  nearly  as  possible  to  zero.  The  mica  acces- 
sory plate  is  then  placed  above  the  comparator,  with  its  center  above  the  darkest 
step,  and  it  is  determined  whether  a  step  becomes  completely  dark,  or  whether  two 
sections  have  equal  illumination. 

For  example,  let  a  mineral  of  2L  (1/2  X)  retardation  be  placed  upon  the  stage  of 
the  microscope  with  its  vibration  directions  at  right  angles  to  those  of  the  compara- 
tor. The  two  will  compensate  when  the  second  step  of  the  comparator  is  placed 
above  it.  If  the  mineral  be  considered  negative  and  the  wedge  positive,  the  first 
step  will  equal  iL  —  2L=  —iL,  and  the  third  ^L  —  2L=iL  (Fig.  462).  That  is,  the 
steps  on  either  side  of  the  one  which  becomes  dark  are  equally  illuminated. 

If  the  mineral  differs  by  half  a  Levy  from  any  step  of  the  comparator,  this  is 
shown  by  equal  illumination  of  two  adjacent  steps  (Fig.  463).  Thus  a  2  1/2  L 
mineral  will  make  the  first  step  i  L  —  2  1/2  L=  — i  1/2  L,  the  second  2L  —  2  1/2  L 
=  —i/2  L,  the  third  3  L  —  2  1/2  L=i/2  L. 

So  far  without  the  mica  accessory  plate.  If  this  be  placed  above  the  compara- 
tor, the. values  in  Levys  of  the  first  three  steps  with  no  mineral  on  the  stage  will 
be  as  shown  in  Fig.  464. 

Suppose  a  mineral,  differing  by  1/4  L  from  one  of  the  steps,  for  example  the 
second  (therefore  =2  1/4  L),  be  placed  on  the  stage  with  its  vibration  directions 


lOp.  «'/.,  1897-8. 


382 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  308 


at  right  angles  to  those  of  the  comparator.     The  values  become  i  1/4  L  —  2  1/4  L  = 

—  iL,  2  1/4 L  —  2  1/4 L  =  o,  $L  —  2  1/4 L  =  3/4 Z,,  iL  — 2  1/4!,=  —i  1/4  L,i  3/4 L 

—  2  1/4  L=  —  i/2  L,  2  3/4^  —  2  1/4  L=  1/2  Z,.    These  are  shown  in  Fig.  465.     A 
difference  of  3/4  Z,,  of  course,  will  be  but  a  difference  of  1/4  I,  in  the  opposite 
direction.    This  is  shown  in  Fig.  466,  where  a  i  3/4  L  mineral  was  used. 

A  difference  of  1/8  L  is  shown  in  Fig.  467,  with  a  2  i/S  L  mineral.     Here  no 
section  is  reduced  to  zero  but  two  spaces  are  uniformly  lighted. 


.± 

2 

1 
2 

•4 

3 

0 

i 

4 

FIG.  464. — No  mineral  on  the 
stage.  Mica  accessory  plate 
overlying  the  first  three  steps  of 
the  compensator. 


FIG.  465. — A  difference  of 


FIG.  466. — A  difference  of 
3/4  L. 


A  2  3/8  L  mineral  giving  a  3/8  L  difference  is  shown  in  Fig.  468. 
A  i  5/8  L  mineral  giving  a  5/8  L  difference  is  shown  in  Fig.  469. 
A  i  7/8  L  mineral  giving  a  7/8  L  difference  is  shown  in  Fig.  470. 
These  different  cases  may  be  summarized  thus.     Calling  the  upper  row  of  sec- 
tions positive  and  the  lower  row  negative,  we  have: 

One  Levy  difference  is  shown  by  darkness  of  a  full  step  (Fig.  462). 
One-half  Levy  difference  is  shown  by  equal  illumination  of  two  adjacent  steps 
(Fig.  463). 


FIG.  467.  FIG.  468.  FIG.  469. 

FIGS.  467  to  470. — Differences  of  1/8,  3/8,  5/8  and  7/8  L. 


FIG.  470. 


1/4  L  difference,  by  darkness  of  section+2  and  equal  illumination  of —  2  and  — 3. 

3/4  L  difference  by  darkness  of  section  —  2  and  equal  illumination  of  -f- 1  and+2. 

1/8  L  difference  by  equal  illumination  of  +i  and  +3. 

3/8  L  difference  by  equal  illumination  of  +3  and  —2. 

5/8  L  difference  by  equal  illumination  of  +2  and  —  i. 

7/8  L  difference  by  equal  illumination  of  —  i  and  —3. 

It  is  thus  possible  to  get  definite  reductions  to  1/8  L  (1/32  X),  and  one  may  even 
clearly  see  a  change  of  1/64  X,  in  which  case  the  illuminations  of  the  proper  sections 
are  not  quite  equal. 

It  is  very  important,  in  making  these  delicate  measurements,  to  be  sure  that 
the  steps  of  the  comparator  and  the  mica  accession  plate  are  truly  1/4  X  and 
1/16  X,  which  may  be  done  by  comparing  two  mica  wedges  by  compensation. 


ART.  314]    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE  383 

If  the  original  comparator  were  made  in  steps  of  0.2  X  instead  of  0.25  X,  the  divi- 
sions could  be  written  in  simple  decimals. 

309.  Salomon's  Method  for  Computing  the  Value  of  co— e  in  Uniaxial 
Minerals  (1869). — In  connection  with  the  determination  of  the  refractive 
indices  of  unknown  minerals  by  means  of  comparison  with  those  of  quartz, 
Salomon  suggested  a  method  by  which  the  order  of  birefringence  and  the 
thickness  of  section  of  uniaxial  minerals  may  be  accurately  computed.     It 
is  described  in  full  in  Art.  242.     The  method  may  be  applied,  not  only  to 
quartz,  but  to  any  uniaxial  mineral. 

310.  Wallerant's  Method  for  Measuring  Slight  Double  Refraction  (1899). 
— Interference  colors,  lower  than  first  order  yellowy  maybe  hard  to  distinguish. 
Wallerant1  devised  a  method  by  which  the  color  may  be  doubled,  and,  in 
consequence,  more  easily  measured.     He  placed  a  horizontal  mirror  under 
the  section,  and  reflected  the  light  along  the  axis  of  the  microscope  by  means 
of  a  small  glass  plate  inserted  at  an  angle  of  45°  in  the  opening  from  which 
the  Bertrand  lens  had  been  removed.     These  reflected  rays  were  polarized 
by  the  analyzer,  passed  through  the  crystal  twice,  once  before  and  once  after 
reflection  from  the  mirror,  and  returned  through  the  analyzer  to  the  eye. 
The  color  seen,  therefore,  was  the  same  as  that  of  a  section  twice  as  thick 
between  parallel  nicols,  and  could  be  determined  readily  by  the  Michel-Levy 
comparator  or  any  other  method. 

311.  Nikitin's  Method  (1900). — Von  Fedorow's  universal  stage  maybe 
used  in  the  determination  of  birefringence,  as  was  shown  by  von  Fedorow 
and  by  Nikitin.     The  method  is  described  in  Art.  443. 

312.  Joly's  Method  (1901). — A  method,  very  similar  to  that  of  Wallerant, 
was  used  by  Joly.2     Instead  of  reflecting  the  light  twice  through  the  analyzer 
he  used  a  third  nicol  outside  the  tube.     After  passing  through  this  nicoi, 
the  light  was  reflected,  by  means  of  a  prism  above  the  objective,  to  a  polished 
speculum  metal  or  silver  mirror  beneath  the  thin  section.     In  this  way, 
double  the  interference  color  was  seen  between  crossed  nicols.     Joly  suggested 
placing  the  section  with  cover-glass  down  so  that  the  rock  slice  is  as  near  as 
possible  to  the  mirror.     By  this  method  it  is  possible,  also,  to  double  the 
colors  of  minerals  with  slight  pleochroism. 

313.  Wright  Combination  Wedge  (1901). — The  Wright  quartz-gypsum 
wedge  has  been  described  in  Art.  297.     The  method  for  determining  the 
double  refraction  is  the  same  as  with  the  ordinary  quartz  wedge. 

314.  Evans  Simple  Quartz  Wedge  (1905). — Evans,3  in  1905,  proposed 

1  Fred.  Wallerant:    Note  sur  la  mesure  des  birefringences  des  miner aux  en  lames  minces. 
Bull.  soc.  min.  France,  XX  (1897),  172-3. 

2  J.  Joly:    On  an  improved  method  of  identifying  crystals  in  rock-sections  by  the  use  of 
birefringence.     Proc.  Roy.  Soc.,  Dublin,  IX  (1901),  485-494. 

3  John  W.  Evans:    On  some  new  forms  of  quartz-wedges  and  their  uses.     Mineralog.  Mag., 
XIV  (1905),  87-92. 


384  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  315 

two  new  quartz  wedges.  The  first  was  simply  an  ordinary  quartz  wedge  of 
larger  angle  than  usual.  It  was  about  11/2  mm.  thick  at  the  thick  end, 
and  gave  twenty-eight  orders  of  interference  colors.  On  the  upper  surface 
a  scale  was  engraved,  the  relative  retardation  of  two  adjacent  divisions  differ- 
ing by  iooo/i,u.  The  calibration  was  effected  in  sodium  light,  first  with 
crossed  nicols  to  obtain  the  dark  bands  corresponding  to  even  half-wave 
lengths,  and  then  between  parallel  nicols  for  the  odd  half-wave 
lengths. 

315.  Evans  Double  Quartz  Wedge  (1905).  —  Evan's  second  wedge  consists 
of  two  quartz  wedges  placed  close  together,  one  with  its  length  parallel  to 
the  optic  axis  of  the  quartz,  the  other  with  its  breadth  parallel  to  the  same 
direction.    The  two  wedges  are  ground  down  to  the  same  slope  whereby, 
since  the  vibration  directions  are  at  right  angles  to  each  other,  they  ex- 
tinguish simultaneously  between  crossed  nicols.     In  the  45°  position  the 
bands  of  color  extending  across  the  wedges  are  the  same  in  both.     If,  however, 
a  mineral  is  placed  on  the  stage  of  the  microscope  and  it  is  rotated  45°  off 
extinction,  one  wedge  will  show  the  black  compensation  bar  while  the  other 
will  show  colors  of  increased  retardation,  double  that  of  the  crystal  plate. 

This  wedge  is  very  convenient,  since  it  is  not  necessary  to  experiment 
first  45°  to  the  right  and  then  to  the  left  to  get  the  mineral  into  position  for 
compensation.  In  determining  extinction  angles  it  is  to  be  noted  that  at 
extinction  the  bands  pass  across  the  two  without  break. 

316.  Seidentopf  Quartz  Wedge  Compensator  (1906).  —  The  Seidentopf1 
compensator  (Fig.  471)  consists  of  a  Ramsden  ocular  in  whose  focal  plane 

there  may  be  placed  movable  quartz 
wedges.  To  reduce  the  length, 
three  are  provided,  one  from  o  to 
the  2d  order,  another  from  the  2d  to 
the  8th,  and  a  third  from  the  8th 
to  the  39th.  The  wedges  are  sim- 
ple except  the  first,  which  consists  of 
a  superimposed  pair  with  their  optic 
axes  at  right  angles  to  each  other, 
thus  giving,  at  a  certain  point,  ex- 

FIG.  471.—  Seidentopf  quartz-  wedge  compensator,     act    Compensation.       The    Upper    SUr- 
1/2  natural  size.     (Fuess.)  faces    ^    graduated    SQ  that   the  re_ 

tardation  may  be  read  directly  from  the  scale;  the  first  and  second  to 
o.i/x  and  estimated  to  o.oiju,  the  third  to  0.2/4  and  estimated  to 


1  H.   Seidentopf:    Mikroskop-Okular  mil  Quarzkeil-Kompensator.     Centralbl.   f.  Min. 
etc.,  1906,  745-746. 


ART.  318]    DETERMINATION  OF  THE  ORDER  OF  BIREFRINGENCE  385 

317.  Wright  Double  Combination  Wedge  (1908). — The  Wright1  double 
combination  wedge  (Fig.  472)  is  made  by  cutting  in  halves,  longitudinally, 
a  single  combination  wedge  whose  line  of  compensation  is  at  the  middle,  and 
rotating  one-half  through  180°  in  azimuth.  By  this  means  the  wedge  is 
divided  into  four  parts.  In  each  half,  like  in  the  single  wedge,  the  retardation 
effect  of  the  wedge  predominates  at  one  end  and  the  underlying  plate  at 
the  other.  Since  their  vibration  directions  lie  at  right  angles  to  each  other, 
this  produces,  in  the  double  wedge,  retardation  as  shown  in  the  figure,  the 
predominating  vibration  directions  being  shown  by  the  shading. 


FIG.  472.  —  Wright  double  combination  FIG.  473.  —  Nikitin  compensator.     (Fuess.) 

wedge. 

318.  Nikitin  Quartz  Compensator  (1910).  —  Another  attachment  for 
determining  low  values  of  birefringence  is  that  proposed  by  Nikitin.2  It 
consists  (Fig.  473)  of  a  plate  of  quartz  cut  so  that  its  normal  makes  an  angle 
of  25°  with  the  optic  axis,  and  is  inserted  in  a  carrier  which  fits  into  the  slot 
above  the  objective.  The  quartz  plate  may  be  rotated  by  means  of  a  milled 
head,  and  is  so  arranged  that  the  optic  axis  moves  in  a  plane  at  right  angles 
to  the  pivot.  Upon  inserting  this  accessory  in  the  microscope  between 
crossed  nicols,  the  stage  will  appear  dark  when  the  pointer  is  at  o.  A 
rotation  of  any  amount  from  this  position  will  produce  an  interference 
color,  violet  of  the  first  order  appearing,  as  shown  in  the  figure,  when  the 
scale  indicates  60°,  in  which  position  the  optic  axis  of  the  quartz  is  inclined 
35°  to  the  axis  of  the  microscope.  Further  rotation  will  produce  higher 
colors. 

In  the  instrument  show^n,  the  quartz  plate  has  a  thickness  of  0.07  mm., 
and  all  the  colors  of  the  first  order  may  be  obtained.  By  using  minerals  of 
greater  birefringence  for  the  plate,  it  would  be  possible  to  increase  the  orders 
of  colors.  The  maximum  error  of  observation  with  this  instrument  is  not 
over 


1  Fred  Eugene  Wright:    On  the  measurement  of  extinction  angles  in  the  thin  section. 
Amer.  Jour.  Sci.,  XXVI  (1908),  370. 

Idem:  The  metlwds  of  petrographic-microscopic  research.     Washington,  1911,  134-135. 
The  illustrations  of  this  wedge  given  in  the  above  publications  are  incorrect,  and  differ 
from  the  letter-press  descriptions. 

2  W.   Xikitin:    Drehbarer  Compensator  fiir  Mikroskope.     Zeitschr.   f.   Kryst.,  XLVII 
(1910),  378-379. 

25 


CHAPTER  XXVI 
DETERMINATION  OF  VERY  SLIGHT  DOUBLE  REFRACTION 

319.  Blot's  Sensitive  Violet  (1813). — For  the  determination  of  very  slight 
double  refraction,  the  usual  accessory  employed  is  the  sensitive  violet.     By 
examining  the  scale  of  interference  colors,1  it  may  be  seen  that  a  very  slight 
retardation  produces  a  decided  change  in  color,  both  with  crossed  nicols, 
giving  a  retardation  of  575///*>  °r  with  parallel  nicols,  giving  a  retardation 
of  2&ifjLn.     The  method  of  observation  is  to  insert,  above  the   apparently 
isotropic  mineral,  a  unit  retardation  plate,  and  note  whether  there  is  a  change 
in  the  interference  color  when  the  stage  is  rotated. 

Besides  the  gypsum  plate  of  Biot  already  described2  the  following  have 
been  used: 

320.  Biot  Quartz  Plate  (1813). — The  quartz  plate  proposed  by  Biot3 
consists  of  a  section  of  quartz  cut  at  right  angles  to  the  optic  axis.     Owing 
to  the  thinness  of  the  plate,  the  rotary  polarization  is  not  noticeable,  and  the 
section  will  appear  dark  between  crossed  nicols.     If,  however,  a  mineral 

showing  slight  double  refraction  is  placed  on  the  stage, 
the  plate  will  appear  colored.  This  plate  was  further 
developed  by  Klein  (Art.  324). 

321.  Savart  Plate  (before  1835). — An  extremely 
delicate  test  for  small  amounts  of  polarized  light  is 
Savart's4  polariscope.  This  consists  of  two  plates  of 
quartz  or  calcite,  cut  at  4^°  to  the  optic  axis,  superposed 

FIG.  474-— Savart  plate.  .     .  . 

with  their  principal  sections  at  right  angles  to  each 
other,  and  cemented  with  Canada  balsam.  When  such  a  plate  is 
mounted  in  front  of  an  analyzer,  nothing  is  seen,  but  if  the  entering 
light  be  ever  so  slightly  polarized,  parallel  bands  bisecting  the  angle 
between  the  principal  sections  of  the  plate  immediately  appear  (Fig. 
474).  These  are  known  as  Savart's  bands,  and  increase  in  strength  as 
the  plane  of  polarization  approaches  the  direction  of  the  bands  them- 

1  Arts.  276-277. 

2  Art.  294. 

3  J.  B.  Biot:  Memoir e  sur  un  nouveaux  genre  d' oscillation  que  les  molecules  de  la  lumiere 
eprou-oent  en  travetsant  certains  cristaux.  Lu  a  1'Institute,  3  Nov.  1813.     Mem.  Acad.  France. 
Ann6e  1812,  XIII,  pt  i  (1814),  1-371. 

Idem:  Precis  elementaire  de  physique  experimentale.     Paris,  1824,  II,  572. 

4  Original  reference  not  found.     It  was  in  use  as  early  as  1835. 

386 


ART.  324]     DETERMINATION  OF  SLIGHT  DOUBLE  REFRACTION  387 

selves,  that  is,  bisects  the  angle  between  the  principal  sections  of  the 
component  plates.  The  two  plates  should  be  of  exactly  the  same  thickness, 
and  are  best  prepared  by  using  the  two  halves  of  a  single  preparation. 

The  instrument  is  extremely  sensitive  and  is  capable  of  detecting  the 
polarization  of  light  reflected  from  the  sky.  It  might  be  used  with  advantage 
for  some  purposes  in  petrographic  work. 

322.  Soleil  Bi -quartz  Plate  (1845). — The  Soleil1  double  quartz  plate  is 
based  on  the  principle  of  the  Biot  quartz  plate.     It  consists  of  two  adjacent, 
equally  thick,  right-  and  left-handed  quartz  plates,  cut  accurately  at  right 
angles  to  the  optic  axis.     This  plate,  as  such,  is  not  much  used  in  petrographic 
work,  but  in  many  saccharimeters  it  is  the  testing  plate.     If  rotary  polariza- 
tion occurs  in  the  substance  under  examination,  the  rays  are  turned  in  one 
direction  by  one  half  of  the  plate  and  in  the  other  by  the  other  half,  conse- 
quently different  interference  colors  appear.     With  no  mineral  plate  on  the 
stage  and  with  parallel  nicols,  the  two  halves  appear  equally  illuminated 
when  the  rotation  is  exactly  90°  or  180°,  monochromatic  light  being  used. 
This  occurs  for  sodium  light  when  the  quartz  is  4.1  mm.  thick,2  since  i  mm. 
of  quartz  produces  a  rotation  of  21.7°.     With  a  thickness  of  8.25  mm.  the 
plane  of  polarization  will  be  rotated  180°  and  parallel  nicols  will  give  the 
violet  "teinte  sensible." 

323.  Bravais  Twinned  Mica  Plate  (1851). — This  is  described  in  full  in 
Art.  335. 

324.  Klein  Quartz  Plate  (1874;. — If  a  thick  plate  of  quartz,  cut  at  right 
angles  to  the  optic  axis,  is  inserted  between  crossed  nicols,  there  will  appear 
an  interference  color  which  will  increase  or  decrease  in  the  scale  upon  rotating 
the  upper  nicol,  the  color  depending  upon  the  amount  of  rotation.     If  a 
mineral  plate  is  placed  between  the  quartz  plate  and  the  polarizer,  the  result- 
ing interference  color  will  be  a  combination  of  the  two.     Klein3  took  advan- 
tage of  this  power  of  quartz  and  constructed  a  plate,  3.75  mm.  in  thickness, 
which  is  very  useful  for  detecting  slight  double  refraction,  slight  differences 

1  Henri  Soleil:  Note  sur  un  moyen  de  faciliter  les  experiences  de  polarisation  rotatoire. 
Comptes  Rendus,  XX  (1845),  1805-1808. 

Idem:  Xouvel  appareil  propre  a  la  meswe  des  deviations  dans  les  experiences  de  polarisa- 
tion rotatoire.  Ibidem,  XXI  (1845),  426-430. 

Idem :  Xote  sur  un  perfectionnement  apporte  au  poinlagc  du  saccharimetre.  Ibidem,  XXIV 
(1847),  973-975- 

M.  M.  Arago,  Regnault  et  Babinet:  Rapport  sur  le  saccharimetre  de  M.  Soliel.  Ibidem, 
XXVI  (1848),  162-168. 

Jules  Duboscq  et  Henri  Soleil :  Note  sur  un  nouveau  compensateur  pour  le  saccharimetre. 
Ibidem,  XXXI  (1850),  248-250. 

H.  Landolt:  Das  optische  Drehungsvermogen.     Braunschweig,  2  Aufl.,  1898,  295. 

2  Cf.  Art.  78. 

3  Carl  Klein:  Mineralogisclic  Mittheilungen  IV.     Xeues  Jahrb.,  1874,  9. 


388  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  325 

in  the  extinction  angles  of  twinned  plates,  and  polarization  of  light  reflected 
from  opaque,  metallic,  anisotropic  surfaces.1  Very  slight  differences  in  the 
orientation  of  the  vibration  directions  produce  different  interference  colors, 
no  matter  what  may  be  the  amount  of  the  rotation  of  the  analyzer.  For 
colorless  minerals,  the  most  sensitive  tint  is  the  violet  teinte  sensible,  which 
is  produced  with  the  indicated  thickness  of  plate  when  the  nicols  are  parallel. 
For  a  colored  mineral  the  tone  to  which  it  is  most  sensitive  should  be  selected. 
This  plate  is  sometimes  called  the  Biot-Klein  plate  (Cf.  Art.  320). 

325.  Bertrand  Ocular  (1877). — This  is  described  in  Article  339. 

326.  Calderon  Ocular  (1878). — This  is  described  in  Article  340. 

327.  Traube  Bi-mica  Plate  (1898). — This  is  described  in  Article  343. 

328.  Brace  Half -shade  Elliptical  Polarizer  and  Compensator  (1904).— 
Brace2  made  use  of  an  extremely  thin  mica  flake,  as  thin  as  0.00017  mm-> 
which  he  inserted  in  the  focal  plane  of  the  ocular,  covering  only  one-half  the 
field.     He  claims  that  this  apparatus  is  two  hundred  times  as  sensitive  as 
the  Bravais  plate,  and  is  capable  of  detecting  retardations  of  6.io-5X. 

329.  Sommerfeldt  Twinned  Gypsum  Plate  (1907). — This   is  described 
in  Article' 345. 

330.  Konigsberger  Ocular  (1908). — Konigsberger3  constructed  an  ocular 
in  which  the  sensitive  plate,  is  composed  of  four  mica  plates,  as  thin  as 
possible,  "crossed  in  pairs,"  with  vibration  directions  at  45°  to  those  of 
polarizer  and  analyzer.     The  apparatus  is  sensitive  to  a  difference  of  i.io~4X, 
and  by  it  the  double  refraction  induced  in  a  piece  of  glass  when  lightly 
pressed  between  two  fingers  may  be  detected.     As  made  by  Fuess,  the  mica 
plate  in  the  Konigsberger  ocular  consists  of  two  pieces  only. 

331.  Half -shade  Plates. — To  determine  the  rotating  power  of  a  substance 
by    monochromatic   light,    half-shade   plates    are   generally    used.     These 
consist  of  transparent  plates  so  cut  that  at  a  certain  position  of  the  analyzer, 
depending  upon  the  rotating  power  of  the  substance  under  examination,  a 
uniform  darkening  of  the  plate  takes  place.     On  account  of  the  half  light 
transmitted  at  this  point,  such  devices  are  called  half-shade  plates.4     In 

1  E.  A.  Wulfing:  Ueber  die  empfindlichen  Farben  und  uber  ihre  Anwendung  bei  der  Erken- 
nung  schwach  doppelbrechender  Medien.    Sitzb.  Akad.  Wiss.  Heidelberg,  1910,  24te  Abhandl., 
pp.  16. 

2  D.  B.  Brace:  A  half -shade  elliptical  polarizer  and  compensator.     Physical  Review, 
XVIII  (1904),  70-88.     See  also  Phil.  Mag.,  VII  (1904),  323. 

3Joh.  Konigsberger:  Vorrichtung  zur  Erkennung .  und  Messung  geringster  Doppelbre- 
chung.  Centralbl.  f.  Min.,  etc.,  1908,  729-730. 

4  Halbschattenapparate,    polarimetres  a  penombre. 

For  principles  of  construction  of  half-shade  plates  see  H.  Landolt:  Die  opptische  Dreh- 
ungsvermogen,  Braunschweig,  1898,  300-302. 


ART.  331]      DETEKMfXATIOX  OF  SLIGHT  DOUBLE  REFRACTIOX  389 

petrographic  work  few  of  these  instruments  are  used.  They  are  widely 
used,  however,  to  determine  the  rotating  power  of  liquids,  as  in  saccha- 
rimeters  and  polarimeters. 

The  first  half-shade  plate  was  probably  constructed  by  Jellet1  in  1860, 
although  the  principle  had  been  used  earlier  by  Bravais  and  others.  Among 
half-shade  apparatus  are  those  of  Laurent,2  Lippich,3  Lommel,4  Glan,5 
Landolt,6  Wiedemann,7  Lummer,8  Mace  de  Lepinay,9  Brace,10  Naka- 
mura,11  and  Wright.12  The  reader  is  referred  to  the  original  literature. 
The  instrument  of  Brace  is  described  in  Art.  328;  those  of  Wiedemann, 
Mace  de  Lepinay,  and  Wright  in  the  next  chapter. 

1  Rev.  Prof.  Jellett:     On  a  new  instrument  for  determining  the  plane  of  polarization. 
Kept.  Brit.  Asso.  Adv.  Sci.,  Trans,  of  the  Sections,  Oxford  meeting,  1860,  13. 

2  L.  Laurent:  Sur  V orientation  precise  de  la  section  principale  des  Nicols,  dans  les  appa- 
reils  de  polarisation.     Comptes  Rendus,  LXXXVI  (1878),  662-664. 

Idem:  Sur  le  saccharimetre  Laurent.     Ibidem,  LXXXIX  (1879),  665-666. 
F.  Lippich:  Ueber  die  Vergleichbarkeit  polarimetrischer  Messungen.     Zeitschr.  f.  Instrum. 
XII  (1892),  333-342. 

H.  Landolt:  Die  oplische  Drehungsvermogen.     Braunschweig,  2  Aufl.,  1898,  308-314. 

3  F.  Lippich:    Zur   Theorie  der  Halbschattenpolarimeter.     Sitzb.    Akad.   Wiss.   Wien, 
XCIX  (ii),  1890,  695. 

Idem:  Ueber  die  Vergleichbarkeit  polarimetrischer  Messungen.  Zeitschr.  f.  Instrum., 
XII  (1892),  333-342. 

Idem:  Lotus,  N.  F.  II  (1880).  * 

Idem:  Ueber  ein  neues  Halbschattenpolarimeter.  Zeitschr.  f.  Instrum.,  II  (1882), 
167-174. 

Idem:  Ueber  eine  Verbesserung  an  Halbschattenpolarisatoien.  Ibidem,  XIV  (1894), 
326-327. 

Idem:  DreitheiUger  Halbschatlen-Poiarisator.  Sitzb.  Akad.  Wiss.  Wien,  CV  (ii  A),  1896, 
317-361. 

O.  Lummer:  Neues  Kontrast-Polarimeter.     Zeitschr.  f.  Instrum.,  XVI  (1896),  209-211. 

4  E.  Lommel:    Neue  Methode  zur  Messung  der  Drehung  der  Polatisationsebene  jiir  die 
Fraunhofer'schen  Linien.    Sitzb.  Akad.  Wiss.  Munchen,  XVIII  (1888),  321-324. 

Idem:  Same  title.     Zeitschr.  f.  Instrum.,  IX  (1889),  227. 

6  Paul  Glan:   Ein  Spektrosaccharimeter.     Wiedem.  Ann.,  XLIII  (1891),  441-448. 

6  H.  Landolt:    Ueber  eine  -oerdnderte  Form  des   Polarisationsapparates  fur  chemische 
Zwecke.     Ber.  deutsch.  chem.  Gesell.,  XXVIII  (1895),  3102. 

7  See  Article  342. 

8  O.  Lummer:    Ueber  ein  neues  Halbschatienprinzip.     Zeitschr.  f.  Instrum.,  XV  (1895), 
293-294. 

9  See  Article  344.2 

10  See  Article  328. 

11  S.  Nakamura:  Ueber  einen  Quarzhalbschattenapparat.     Centralbl.  f.  Min.,  etc.,  1905 
267-279. 

12  See  Article  347. 


CHAPTER  XXVII 


PRACTICAL    METHODS    FOR  THE  DETERMINATION   OF  EX- 
TINCTION ANGLES 

332.  Relation  of  the  Optical  Ellipsoid  to  the  Crystallographic  Axes. 
Parallel  and  Inclined  Extinction. — It  has  been  demonstrated  geometrically1 
and  analytically2  that  when  the  principal  vibration  directions  of  a  crystal 
correspond  in  direction  with  the  principal  planes  of  the  nicol  prisms,  the 
light  is  extinguished  and  the  field  becomes  dark.  This,  of  course,  occurs 
four  times  on  rotating  the  stage  through  360°,  and  these  positions  are  called 
the  positions  of  extinction.  We  have  seen,  also,  that  in  isometric  crystals 

the  ease  of  vibration  is  the  same  in  every  direc- 
tion, consequently  the  field  remains  dark  be- 
tween crossed  nicols  during  a  complete  rotation 
of  the  stage.  In  uniaxial  crystals  the  vibration 
ease  is  the  same  in  every  direction  in  basal  sec- 
tions, consequently  these  likewise  remain  dark 
on  rotating  the  stage.  In  sections  of  uniaxial 
minerals  at  right  angles  to  the  base,  double  re- 
fraction occurs,  and  the  field  darkens  only  when 
the  trace  of  the  basal  plane  and  the  direction  at 
right  angles  to  it  are  parallel  to  the  cross-hairs. 
Since  in  uniaxial  crystals  the  principal  vibration 
axes  are  parallel  to  the  crystallographic  axes, 
this  extinction  will  take  place  when  crystallo- 
graphic c  is  parallel  to  the  cross-hairs.  In  sec- 
tions intermediate  between  the  basal  section 
and  the  section  containing  crystallographic  c, 

the  field  will  likewise  become  dark  four  times,  namely,  when  the  trace  of  the 
plane  of  the  base  or  the  one  containing  crystallographic  c  is  parallel  to  the 
cross-hairs.  In  orthorhombic  crystals  (Fig.  475)  the  vibration  axes,  like- 
wise, coincide  with  the  crystallographic  axes,  consequently  when  traces 
of  the  planes  containing  these  lines  are  parallel  to  the  cross-hairs,  the 
field  becomes  dark.  Unlike  uniaxial  crystals,  the  basal  plane  here  extin- 
guishes four  times. 

We  thus  see  that  in  tetragonal,  hexagonal,  and  orthorhombic  crystals  the 


FIG.  475. — Orthorhombic  sys- 
tem. Relation  between  crystal- 
lographic lines  and  the  optical 
ellipsoid. 


1  Art.  283. 

2  Arts.  285  and  287. 


390 


ART.  332] 


DETERMINATION  OF  EXTINCTION  ANGLES 


391 


extinction  lines  are  parallel  to  the  crystallographic  axes.  Now  in  most 
crystals  there  are  cleavage  lines  which  are  parallel  to  the  crystallographic 
axes,  and  when  these  lines  lie  parallel  to  the  cross-hairs,  the  vibration  direc- 
tions also  lie  parallel,  and  the  field  becomes  dark.  Such  extinctions  are  said 
to  be  parallel  (Fig.  476).  The  cross-hairs  may  correspond  to  a,  b,  or  c,  or  to 


PIG.  476. — Parallel  extinction. 


PIG.  477. — Symmetrical  extinction. 


some  intermediate  vibration  direction,  but  we  cannot  determine  which, 
unless  the  orientation  of  the  crystal  is  known.  We  can  simply  determine 
that  the  vibrations  in  one  direction  are  faster  than  in  the  other.  The  relation 
of  the  extinction  lines  to  prismatic  cleavage  in  uniaxial  or  orthorhombic 
crystals  will,  in  certain  sections,  be  symmetrical  (Fig.  477).  In  other  sections, 


4- 

100 


\ 


FIG.  478.  PIG.  479.  FIG.  480. 

PIGS.  478  and  479. — Extinction  angles  in  a  monoclinic  crystal. 
FIG.  480. — Augite  showing  angles  of  extinction  on  the  different  faces  of  a  zone. 

however,  these  extinctions  will  appear  parallel,  and  the  mineral  is  said  to 
have  parallel  extinction. 

In  monoclinic  crystals  there  is  but  one  plane  of  symmetry,  and  the 
optical  ellipsoid  will  have  but  a  single  axis  coinciding  with  a  crystallo- 
graphic axis,  namely,  the  one  at  right  angles  to  this  plane,  or  crystallographic 
b  (Figs.  478-479).  The  other  axes  will  lie  anywhere  in  the  plane  of  a  and  c. 
In  general,  neither  corresponds  with  a  vibration  axis,  but  it  is  possible,  of 


392  MANUAL  OF  PETROGRAPIIIC  METHODS  [ART.  333 

course,  for  one  to  do  so,  but  not  for  both,  since  the  angle  between  a  and  c 
is  not  a  right  angle,  and  that  between  the  vibration  axes  is.  Extinction  in 
such  crystals  is  said  to  be  inclined,  although  in  sections  in  the  100-001  zone 
(Figs.  479-480)  it  will  appear  parallel. 

In  triclinic  crystals  the  vibration  axes  do  not  coincide  with  the  crystallo- 
graphic  axes,  although  in  special  cases,  of  course,  a  single  one  may  do  so. 

The  angle  between  the  extinction  direction  and  a  crystallographic  axis, 
shown  by  cleavage  lines  or  crystal  edges,  is  called  the  extinction  angle  of  the 
face.  Thus  in  Fig.  480,  which  is  a  crystal  of  augite,  the  extinction  angle  on 
oio  is  45°,  on  no  it  is  36°,  and  on  100  it  is  o°.  The  maximum  extinction 
angle  between  any  crystallographic  axis  and  the  nearest  vibration  axis  is 
usually  taken 'as  the  extinction  angle  of  the  mineral.  The  relation  be- 
tween the  angles  on  different  faces  will  be  considered  in  full  in  Chapter 
XXVIII. 

333.  Methods  for  Measuring. — Before  making  accurate  determinations 
of  extinction  angles,  it  is  very  essential  that  the  microscope  be  in  adjustment, 
the  principal  planes  of  the  two  nicols  exactly  at  right  angles  to  each  other  and 
parallel  to  the  cross-hairs.  If  special  oculars  are  used,  these,  also,  should 
be  tested  to  see  that  their  cross  lines  are  parallel  to  the  principal  planes  of  the 
nicols.  For  the  methods  of  adjustment  see  Arts.  199-202. 

Except  for  extremely  accurate  measurements,  extinction  angles  are 
determined  by  white  light.  If  a  crystal  is  to  be  measured,  oriented  sections 
should  be  cut  and  the  maximum  extinction  angle  determined.  With  random 
sections,  such  as  occur  in  a  rock  slice,  many  should  be  examined,  and  the 
maximum  angle  taken  as  the  extinction  angle  of  the  crystal,  or  a  universal 
stage  should  be  used  and  the  angle  determined  by  the  methods  given  in 
Art.  440. 

The  usual  method  of  determining  extinction  angles  is  to  rotate  the 
mineral,  between  crossed  nicols,  to  the  position  of  maximum  darkness.  This 
operation  should  be  repeated  half  a  dozen  times,  and  then  agan  the  same 
number  of  times  with  the  stage  rotated  180°  from  its  former  position.  An 
average  of  the  twelve  readings  will  give  the  extinction  angle.  Increasing 
the  number  of  readings  will  decrease  the  error  of  observation.  Thus  Max 
Schuster,1  in  his  determinations  of  the  extinction  angles  of  the  plagioclase 
feldspars,  made  80  readings  on  each  side  of  the  twinning  line,  80  to  deter- 
mine the  position  parallel  to  the  001-010  edge,  then  turned  the  slide 
cover-glass  down  and  made  a  like  number  of  readings ! 

Owing  to  the  fact  that  the  eye  cannot  always  accurately  determine  the 
position  of  maximum  darkness,  especially  in  sections  showing  but  slight 
birefringence,  various  accessories  have  been  devised,  most  of  them  depending 

1  Max  Schuster:  Ueber  die  optische  Orientierung  der Plagioklase.  T.  M.  P.  M.,  Ill  (1881), 
117-284,  in  particular  146-147. 


ART.  335]  DETERMINATION  OF  EXTINCTION  ANGLES  393 

for  their  efficiency  upon  the  teinte  sensible,  or  upon  the  multiplying  effect 
of  twinned  plates  upon  birefringence. 

334.  Unit  Retardation  Plate. — If  a  unit  retardation  plate,  such  as  has 
already  been  fully  described,1  is  placed  above  a  mineral  rotated  to  the  posi- 
tion of  extinction,  the  sensitive  violet  will  appear  just  as  though  no  mineral 
were  upon  the  stage.     If  the  mineral  be  very  slightly  rotated,  however,  the 
very  small  amount  of  retardation  produced  in  the  transmitted  rays  will 
produce  a  decided  difference  in  the  interference  color,  red  in  one  direction 
and  blue  in  the  other.     Applied  to  the  determination  of  extinction  angles 
this  plate  is  serviceable  only  when  the  mineral  is  colorless  and  of  not  too 
high  birefringence.     It  can  be  used,  also,  only  on  isolated  fragments,  on  grains 
adjacent  to  the  Canada  balsam  around  the  periphery  of  a  rock  section,  or 
adjacent  to  an  absolutely  isotropic  mineral;  this  because  it  is  necessary  to 
rotate  the  mineral  until  the  interference  colors  of  field  and  grain  are  exactly 
the  same. 

335.  Bravais  Twinned  Mica  Plate  (1851). — Probably  the  first  twinned 
plate  used  was  that  of  Bravais.2    He  took  a  mica  plate,  1/9  mm.  in  thickness, 
and   thus   having   a  violet   interference   color  and  giving 

exactly  one  wave  length  retardation  by  yellow  light,  and 
cut  it  along  a  line  making  an  angle  of  45°  with  the  prin- 
cipal section  (Fig.  481).  One  part  was  now  turned  through 
1 80°  in  altitude  so  that  the  upper  surface  became  the 
lower,  and  the  parts  were  cemented  on  glass  in  the  position 
shown  in  the  figure.  Between  crossed  nicols  the  two  parts  FlG-  481.— Bravais 
show  the  same  interference  color  if  the  light  strikes  the 
lower  surface  at  right  angles.  If,  however,  a  mineral  plate  is  placed  on 
the  stage  of  the  microscope,  one  half  the  Bravais  plate  will  add  to  its  in- 
terference color  and  the  other  half  will  subtract  from  it,  the  effect  being  to 
show,  between  the  two  halves,  double  the  actual  retardation.  When  the 
mineral  has  been  rotated  until  the  two  parts  of  the  Bravais  plate  are  uni- 
formly colored,  it  is  in  its  position  of  extinction. 

This,  and  all  twinned  plates,  should  be  carefully  tested.  The  angle 
which  each  part  makes  with  the  bisecting  line  must  be  absolutely  the  same, 
otherwise  there  will  be  an  error  of  reading  equal  to  half  the  difference  between 
them. 

1  Art.  294,  supra. 

2  A.  Bravais:  Description  d'un  nouveau  polariscope  et  recherches  sur  les  doubles  refrac- 
tions pen  energiques.     Comptes  Rendus,  XXXII  (1851),  112-116. 

Idem:  D'un  nouveau  polariscope,  et  recherches  sur  les  doubles  refractions  peu  energiques. 
Ann.  chim.  et  phys.,  XLIII  (1855),  129-149. 

Idem:  Beschreibung  eines  neuen  Polariskops  und  Untersuchung  iiber  die  schwa  chen 
Doppelbrechungen.  Pogg.  Ann.,  XCVI  (1855),  395-414. 


394  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  336 

336.  Kobell  Stauroscope  (1855). — The  Kobell1  stauroscope  consists  of  a 
calcite  plate  cut  at  right  angles  to  the  optic  axis  and  inserted  between  the 
mineral  and  the  analyzer.     In  the  original  polariscope  with  which  this  was 
used,  the  polarizing  plate  was  a  black  glass  mirror  and  the  analyzer  a  tour- 
maline plate.     The  instrument  was  so  arranged  that  the  interference  cross 
of  the  calcite  could  be  seen  on  looking  through  the  eye  lens.     With  no  mineral 
on  the  stage,  the  cross  appeared  undisturbed,  but  with  an  anisotropic  mineral 
inserted  in  any  position  except  that  of  extinction,  the  cross  was  more  or  less 
rotated. 

337.  Klein  Quartz  Plate  (1874). — The  Klein  quartz  plate,  described  in 
Art.  324,  may  be  used  to  determine  extinction  angles  in  isolated  mineral 
grains  or  minerals  adjacent  to  isotropic  media,  hence  around  the  periphery 
of  a  rock  section.     The  method  is  similar  to  that  used  with  the  unit  retarda- 
tion plate  except  that  the  upper  nicol  is  rotated  until  the  desired  sensitive 
tint  is  obtained,  and  its  vibration  plane,  therefore,  in  general  is  not  at  right 
angles  to  that  of  the  lower  nicol. 

338.  Bertrand  Ocular  (1877).— In  the  Bertrand2  ocular,  use  is  made  of  the 
rotating  power  of  sections  of  quartz  cut  at  right  angles  to  the  optic  axis. 

It  differs  from  the  Soleil  double  plate  in  that  it  is  made  up 
of  a  double  pair  of  dextro-rotary  and  levo-rotary  quartz 
plates,  instead  of  a  simple  pair.  The  four  pieces,  each  2.5 
mm.  in  thickness,  are  so  placed  that  the  two  right-handed 
and  the  two  left-handed  quartzes  lie  in  opposite  quad- 
rants (Fig.  482).  They  are  so  inserted  in  the  focal  plane  of 
tne  ocular  that  their  separating  lines  are  exactly  parallel  to 
the  vibration  directions  of  the  nicols,  and  thus  serve  as 
cross-hairs.  The  tube  analyzer  is  removed  and  a  cap  nicol  is  placed  above 
the  ocular.  When  the  nicols  are  crossed,  the  four  quadrants  are  of  a  uni- 
form pale  blue  color,  since  the  vibration  directions  of  all  make  the  same 
angle  with  the  vibration  directions  of  the  nicols.  When  a  doubly  refracting 
mineral  section  is  placed  upon  the  stage  of  the  microscope,  the  opposite 
quadrants  add  to  or  subtract  from  its  retardation,  except  when  the  mineral 
is  in  its  position  of  extinction,  and  they  become  differently  colored.  Ex- 
tremely small  variations  from  the  parallel  position  can  thus  be  determined. 
According  to  Wright3  its  sensitiveness  fof  colorless  minerals  is  such  that 

1  Fr.  v.  Kobell:  Optisch-krystallographische  Beobachtungen  und  uber  ein  neues  Polari- 
skop.    Stauroskop.     Pogg.  Ann.,  XCV  (1855),  320-332. 

2  E.  Bertrand:  Vorrichtung  zur  Bestimmung  der  Schwingungsrichtung  doppeltbrechender 
Krystalle  im  Mikroskop.     Zeitschr.  f.  Kryst.,  I  (1877),  69. 

Idem:  De  V application  du  microscope  a  V etude   de   la    mineralogie.     Bull.  soc.   min. 
France,  I  (1878),  22-28,  especially  27. 

Rosenbusch- Wiilfing :  Mikroskopische  Physiographic,  Ii,  1904,  250-251. 

3  Fred.  Eugene  Wright:  The  methods  of  petro graphic-microscopic  research.     Washington, 
1911,  146. 


ART.  341]  DETERMINATION  OF  EXTINCTION  ANGLES  395 

angles  may  be  read  to  between  0.1°  and  0.5°,  and  a  wave  retardation  of 
±  0.005  can  De  recognized. 

Schraf1  suggested  that  the  lenses  in  an  ocular  fitted  with  a  Bertrand 
plate  be  separated  farther  than  usual  in  order  that  its  influence  upon  the 
refraction  be  eliminated,  and  the  resolving  power  of  the  microscope  remain 
the  same.  When  so  made,  the  instrument  may  be  used  as  an  ordinary 
ocular,  the  lines  between  the  four  quadrants  serving  as  cross-hairs. 

339.  Calderon  Plate  (1878). — The  Calderon2  plate,  like  that  of  Bertrand, 
is  used  with  a  cap  nicol.     The  sensitive  plate,  lying  in  the  focal  plane  of  the 
ocular,  consists  of  an  artificial  twin  of  calcite  which  ^ 

is  constructed  by  sawing  a  rhombohedron  of  Ice- 
land spar  along  the  short  diagonal,    removing  a 
wedge-shaped   piece   from   each   cut   plane,    and    < 
cementing  the  remainder  on  these  new  surfaces. 
The  projecting  and  reentrant  angles  are  removed 

,  .     j.  „    .   ,  rv-          o    \   1          •  483-— Calderon  plate. 

by  grinding  to  two  parallel  faces  (Fig.  483),  leaving 

a  plane-parallel  plate  of  such  a  thickness  that  the  interference  color  is  "white 
of  the  higher  orders."  With  crossed  nicols  the  two  halves  appear  alike, 
but  upon  inserting  a  doubly  refracting  mineral  the  two  are  unequally  illu- 
minated except  when  the  mineral  is  in  the  position  of  extinction.  Calderon 
claims  an  accuracy  to  2  minutes. 

An  objection  to  this  plate  is  that  the  formation  of  a  second  image  by  double 
refraction  is  very  annoying.  In  the  oculars  prepared  for  petrographic 
microscopes  the  field  is  generally  small. 

340.  Von  Fedorow's  Method  by  Means  of  the  Universal  Stage  (1892).— 
The  method  for  the  determination  of  the  extinction  angle  of  a  crystal  from 
the  extinction  angle  of  the  section  under  examination,  by  means  of  the  uni- 
versal stage,  is  described  in  Chapters  XXXV  and  XXXVI. 

341.  Wiedemann  Double  Double-Quartz  Wedge.     (Before  1895).— The 

c  Wiedemann3     double     double- 

, 1 _____  quartz   wedge,    though    designed 

for  the  determination   of  rotary 


polarization   for   different   colors, 
may  be  used  for  the  determina- 

FIG.  484. — Wiedemann   double-double  quartz  wedge.          . 

tion  of    extinction    angles.     The 

instrument  consists  of  two  wedges  (Fig.  484),  each  of  which  is  itself  made 
up  of  a  pair  of  dextro-  and  levogyrate  quartz  wedges.     They  are  cut  with 

1  A.  Schraf:  Ueber  die  Verwendung  der  Bertrand' schen  Quarzplatte  zu  mikrostauro- 
skopischen  Beobachtungen.  Zeitschr.  f.  Kryst.  VIII  (1884),  81-82. 

2L.  Calderon:  Ueber  einige  Modificationen  des  Groth' schen  Universalapparates  und 
iiber  eine  neue  Stanroskopvorrichtung.  Zeitschr.  f.  Kryst.,  II  (1878),  68-73. 

3  Gustav  Wiedemann:  Die  Lehre  von  der  Elektricitat,  2  Aufl.,  Ill,  Braunschweig, 
1805,  1051-1052.* 


396 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  342 


their  bases  at  right  angles  to  the  optic  axes,  and  are  so  superposed  that  the 
wedges  of  like  rotation  are  on  the  same  side.  The  amount  of  rotation  is  in- 
creased by  varying  the  thickness  on  the  axis  of  the  microscope. 

342.  Stober  Quartz  Double  Plate  (1897). — Identical  with  the  Bravais, 
except  that  it  is  made  from  two  quartz  plates  cut  parallel  to  the  optic  axis, 
0.064  mm.  thick,  and  giving  violet  of  the  second  order,  is  Stober's1  quartz 
double  plate.     The  quartz  is  cemented  between  two  round  cover  glasses, 
and  the  plate  thus  prepared  is  inserted  as  near  as  possible  to  the  cross-hairs 
in  the  focal  plane  of  an  ocular  and  in  such  a  position  that  the  artificial  twin- 
ning line  is  parallel  to  one  of  them. 

343.  Traube  Bi-mica  Plate  (1898). — Similar  to  the  Calderon  plate,  but 
much  easier  to  construct  since  no  grinding  is  necessary,  is  the  Traube2  bi- 
mica  plate.     Two  rectangular  strips  (Fig.  485)  are  cut  from  a  quarter  undula- 
tion mica  flake  in  such  a  direction  that  their  axial  planes  make  angles  of  3  1/2° 
with  the  long  edges.     The  two  strips  are  cemented  between  glass  so  that  the 
double  extinction  angle  is  7°.     The  complete  plate  is  placed  in  the  focal 
plane  of  the  ocular,  and  is  used  in  the  manner  of  the  Calderon. 


PIG.  485. — Traube  bi-mica  plate. 


FIG.  486. — Mace  de  Lepinay  half-shade  plate. 


344.  Mace  de  Lepinay  Half -shade  Plate  (1900). — The  Mace  de  Lepinay3 
half-shade  plate  is  nothing  more  than  the  half  of  a  Wiedemann  wedge.  It 
consists  of  a  double-quartz  wedge  (Fig.  486),  one  dextrogyrate  and  one 
levogyrate,  cut  at  right  angles  to  the  optic  axis  and  varying  in  thickness 
from  0.06  mm.  to  0.12  mm.  The  base  of  the  wedge  is  'turned  toward  the 
analyzer  and  is  placed  as  close  to  it  as  possible.  The  slanting  surface,  how- 
ever, causes  a  slight  deflection  of  the  light.  Schonrock4  suggested  that  this 

1  F.  Stober:  Ueber  eine  empfindliche    Quarzdoppelplatte.     Zeitschr.    f.   Kryst.,  XXIX 
(1897-9),  22-24. 

2  Hermann   Traube:    Eine  einfache  Glimmer  do  ppdplatie   zu  stauroskopischen    Bestim- 
mung.     Neues  Jahrb.,  1898  (I),  251. 

3L.  Mace  de  Lepinay:  Sur  un  nouvel  Analyseur  a  penombres.  Jour,  de  phys.,  IX 
(1900),  footnote  267,  585-588,  644. 

Idem:  Same  title.     Comptes  Rendus,  CXXXI  (1900),  832-834. 

Idem:  Determination  des  constantes  optiques  du  quartz  pour  la  radiation  verte  du  mercure. 
Leur  application  aux  mesures  d 'epaisseurs  par  la  methode  de  Mouton.  Jour,  de  phys.  IX 
(1900),  644-652. 

4  O.  Schonrock:  Neuer  Halbschattenanalysator.  Zeitschr.  f.  Instrum.,  XXI  (1901), 
QO-93- 


ART.  346] 


DETERMINATION  OF  EXTINCTION  ANGLES 


397 


might  be  overcome  by  using  two  wedges  of  different  thicknesses,  which  makes 
of  it,  however,  a  Wiedemann  double  double-quartz  wedge. 

345.  Sommerfeldt  Twinned  Gypsum  Plate  (1907). — The  cheap  device 
proposed  by  Sommerfeldt1  for  determining  whether  nicols  are  absolutely 
at  right  angles  to  each  other  may  well  be  used  for  the  determination  of  extinc- 
tion angles.  He  used  a  cleavage  plate  of  a  twinned  gypsum  crystal  in  which 
the  trace  of  the  twinning  plane  is  a  straight  line  (Fig.  487).  The  two  indi- 
viduals appear  equally  illuminated  between  crossed  nicols  when  the  twinning 
line  is  parallel. or  at  45°  to  the  cross-hairs. 

If  an  anisotropic  mineral  plate  is  inserted,  the  two  parts  of  the  field  become 
differently  colored  unless  the  mineral  is  exactly  at  extinction. 

The  present  writer  has  used,  for  a  number  of  years,  a  wedge  made  from 
a  twinned  gypsum  crystal.  It  is  cut  with  its  long  direction  parallel  to  the 
twinning  line,  and  shows  colors  from  gray  of  the  first  to  pink  of  the  fourth 
order. 


FIG.  487. — Sommerfeldt  twinned  PIG.  488. — Wright  artificially 

gypsum  plate.  twinned  quartz  plate. 


346.  Wright  Artificially  Twinned  Quartz  Plate  (1908).— The  twinned 
quartz  plate,  suggested  by  Wright,2  is  similar  to  Sommerfeldt's  plate,  but 
is  made  from  quartz.  This  is  cut  parallel  to  c,  and  with  one  lateral  edge 
ground  down  until  it  makes  an  angle  of  from  3°  to  6°  with  this  axis  (Fig. 
488).  The  plate  is  cut  across  transversely,  and  the  two  inclined  edges  are 
cemented  together.  Wright  suggests  that  such  plates  may  be  made  as 
quarter  undulation  plates,  first  order  violet  plates,  or  even  in  the  form  of 
w  edges. 

1  Ernst  Sommerfeldt:    Eine  einfache  Methode  zur  Justierung  der  Nikols  am  miner  alog- 
ischcn  Mikroskop.     Zeitschr.  wiss.  Mikrosk.,  XXIV  (1907),  24-25. 

See  also  Max  Berek:  Die  Bestimmung  von  Ausloschungsrichtnngen  doppellbrechender 
i  mi  k  liver  Krystallplatten  mil  Hilfe  von  Halbschattenvorrichtungen  im  einfarbigen  Lichte. 
Xeues  Jahrb.,  B.B.,  XXXIII  (1912),  583-661. 

2  Fred.  Eugene  Wright:    On  the  measurement  of  extinction  angles.     Amer.  Jour.  Sci., 
XXVI  (1908),  374. 

Idem:  The  methods  of  petro *  graphic-microscopic  research.     Washington,  1911,  136-137. 


398 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  347 


FlG.  489. — Wright  bi-quartz-wedge  plate. 


347.  Wright  Bi-quartz  Wedge  Plate  (1908).— The  Wright1  bi-quartz 
wedge  plate  consists  of  two  quartz  wedges,  one  dextrogyrate  and  one  levogy- 
rate,  each  underlaid  by  a  plane-parallel  quartz  plate  of  opposite  sign  (Fig. 
489),  thus  producing  zero  rotation  where  the  plates  are  of  the  same  thickness, 
much  in  the  manner  of  the  Wright  single  combination  wedge.  This  wedge 

is  inserted  in  the  focal  plane  of  an 
ocular.  It  divides  the  field  into 
halves  of  equal  illumination  when 
the  stage  is  bare  or  when  a  mineral 
placed  thereon  has  its  extinction  di- 
rections parallel  to  the  principal  planes  of  the  nicols.  A  very  slight  rotation 
produces  a  difference  in  the  amount  of  the  illumination  in  the  two  parts, 
the  most  marked  difference  being  found  by  inserting  or  withdrawing  the 
wedge,  more  or  less.  To  avoid  tilting  the  wedge,  and  thus  permitting  light 
to  pass  through  in  a  directkxi  other  than  parallel  to  the  optic  axis,  it  is 
set  in  a  carriage  which  slides  snugly  in  an  ocular  very  similar  in  appear- 
ance to  that  shown  in  Fig.  471. 

JFred.  Eugene  Wright:  On  the  measurement  of  extinction  angles.  Amer.  Jour.  Sci., 
XXVI  (1908),  377- 

Idem:  The  bi-quartz  wedge  plate  applied  to  polarimeters  and  saccharimeters.  Ibidem, 
391-398. 

O.  Schonrock:  Keilformiger  Biquarz  fur  Polarisationsapparate  und  Saccharimeter. 
Zeitschr.  f.  Instrum.,  XXX  (1910),  198-199. 

Fred.  Eugene  Wright:  The  methods  of  petrographic-microscopic  research.  Washington, 
1 91 1,  footnote,  141. 


CHAPTER  XXVIII 


CALCULATION  OF  EXTINCTION  ANGLES  IN  RANDOM 
THIN  SECTIONS 

348.  Zones. — A  zone  has  been  denned  as  being  made  up  of  all  sections 
which  are  parallel  to  the  same  line,  called  the  axis  of  the  zone,  but  not  parallel 
to  each  other.     Thus  the  100,  101,  ooi  faces  lie  in  a  single  zone,  as  do  also 
100,  no,  and  oio. 

We  have  seen  that  the  extinction  angles  in  the  100,  no,  oio  zone  of 
crystals  of  the  monoclinic  system  vary  from  zero  on  100  to  a  definite  value 
on  oio.  In  the  zone  100,  101,  ooi,  the  value  remains  zero  throughout, 
just  as  in  every  zone  of  uniaxial  crystals.  In  triclinic  crystals  the  values 
vary  in  every  zone  from  a  minimum,  exceptionally  zero,  to  a  maximum. 
From  an  examination  of  all  cases,  the  rule  may  be  stated  that  parallel  extinc- 
tion occurs  in  all  tfie  planes  of  a  zone  whose  axis  coincides  with  an  axis  of 
symmetry. 

349.  Calculation  of  Extinction  Angles  for  any  Face  of  the  100-010  Zone  of  a 
Monoclinic  Crystal. — To  determine  the  traces  of  the  vibration  planes  on  any  face 
of  a  crystal,  use  may  be  made  of  Fresnel's  law 

which  states  that  in  any  section  of  a  biaxial 
crystal  (abm,  Fig.  494),  the  direction  of  extinc- 
tion (md)  is  at  the  intersection  of  the  plane  of  the 
section  (abm)  with  the  plane  (mdDM)  bisecting 
the  angle  between  the  two  planes  (mbBM  and 
ma  AM)  containing  the  optic  axes  (MB,  MA) 
and  a  line  at  right  angles  to  the  section  (mM), 

That  the  plane  bisecting  this  angle  is  one  of 
the  vibration  directions,  and  the  plane  at  right 
angles  to  it  is  another,  may  be  proved  very  simply 
by  means  of  a  stereographic  projection.  Let  A 
and  B  (Fig.  490)  be  the  projection  of  the  optic 
axes,  and  P  the  projection  of  the  normal  to  the 
section  upon  which  the  extinction  is  to  be  measured.  Draw  two  circles,  iki' 
and  ik'i'y  polar  to  B  and^l.  They  therefore  represent  the  intersections  of  the 
circular  sections  of  the  optical  ellipsoid  with  the  sphere  of  projection  and  lie 
at  right  angles  to  the  optic  axes  A  and  B.  Draw  k'k,  the  trace  of  the  plane  of 
which  P  represents  the  normal,  and  draw  PB  and  PA,  two  planes  through 
the  points  PA  and  PB  and  the  center  of  the  sphere.  Let  a  and  b  be  the 
points  at  which  the  traces  of  the  planes  PA  and  PB  cut  the  plane  k'k.  Since 
the  traces  of  planes  at  right  angles  to  lines  lie  90°  from  the  piercing  points  of  these 

399 


FIG.  490. 


400 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  349 


lines,  Pb  amd  Bk  are  90°  apart,  therefore  bk  =  go°.  Ak'  and  Pa  are  90°  apart, 
therefore  also  ak'  =  <)o°,  whereby  bk  =  ak'  and  bk'  =  ak.  Since  iki'  and  ik'i'  are  the 
traces  of  the  circular  sections  of  the  optical  ellipsoid,  their  bisecting  plane  id' 
contains  the  bisectrix,  and  the  distance  k'c  must  equal  kc,  whereby  k'c-\-k'b  =  kc+ka, 
and  bc  =  ca.  The  bisecting  plane  PC,  therefore,  passes  through  an  axis  of  the 
ellipsoid  which  determines  one  of  the  vibration  directions.  The  other  is  at  right 
angles  to  the  first  and  is  shown  in  pPp'- 

Returning  to  the  problem  of  determining  the  trace  of  the  vibration  plane  on 
any  face  in  the  100-010  zone  of  a  monoclinic  crystal:  Let  Fig.  491  represent  a  oio 
section  from  such  a  crystal.  This  corresponds  with  its  symmetry  plane.  Let  MC 


FIG.  491. 


FIG.  492. 
FIG.  493. 


FIG.  494. 


be  the  crystallographic  axis,  and  MA  and  MB  the  optic  axes,  then  MD,  which 
bisects  the  angle  BMA,  is  the  acute  bisectrix.  It  therefore  is  a  vibration  and  an 
extinction  direction. 

In  Figs.  492  and  494  let  OA  be  the  trace  of  the  oio  plane,  and  let  it  be  so  placed 
that  the  new  section  plane  Oa,  upon  which  the  extinction  is  to  be  determined,  lies 
at  right  angles  to  the  line  of  sight.  OA,  therefore,  will  form  an  angle  of  <p  (AOa) 
with  Oa,  A  a  being  the  direction  of  the  line  of  sight;  and  the  intercepts  on  the  new 
plane  (Odea,  Figs.  492,  493,  and  494)  will  represent  the  distances  as  they  now  appear. 
In  this  plane  the  vertical  distance  me  (Figs.  493-494)  appears  in  its  true  length 
(MC  =  mc)  since  it  is  parallel  to  the  line  about  which  the  plane  was  rotated.  All 
other  lines  will  appear  foreshortened,  and  all  angles  will  be  of  less  than  natural  size. 

If  we  let  0'  =  the  angle  bmc,  0  =  BMC,  ol  =  amc,  and  a  —  AMC,  we  have,  from 
the  figure, 

be        be        be       BC 


Likewise  tan  a'  =  cos  <?  tan  a. 

If  dmc,  the  extinction  on  the  new  plane,  =  7,  and 


=  bmd+dma=2f 


p'+a  =bma=2v. 
Multiply  (3)  by  two,  and  unite  with  (4) 


(2) 

(3) 
(4) 


27  =£- 


(5) 


ART.  349] 


CALCULATION  OF  EXTINCTION  ANGLES 


401 


By  trigonometry,  from  the  equation  of  the  tangent  of  the  difference  between  two 

angles  we  have 

tan  /?'  —  tan  a 


(6) 


Substitute  values  from  (i),  (2),  and  (5), 

cos  <f>  tan  B  —  cos  <p  tan  a 
tan  27  = 


But 

Substitute  these  values  in  (6), 


i-ftan  a  tan  /?  cos2  <f> 
/3=F+r,  and  «=Z7-I\ 


cos  <p  (tan  (F+T)-tan  (F-T)     i 

tan  2  v  =  — 


(7) 

From  this  equation  the  extinction  angle  (7)  in  any  section  in  the  prismatic  zone  of 
monoclinic  crystals  whose  value  for  V  is  less  than  T,  may  be  determined  when  the 
inclination  (<p)  of  the  section  to  the  oio  plane,  the  maximum  extinction  angle  (T).  and 
the  optic  axial  angle  (2V)  are  known. 

Equation  7  may  be  written  in  terms  of  cotangents — 

cot  (F+T)cos(F-T)+cos2  * 
cos  dcot  (F-T)-cot(F+r)]' 
If  the  extinction  angle  is  greater  than  F(that  is  F>  F),  equations  (4)  and  (5) 
become 


cot  27  = 


and  2v=p'+a.  (10) 

Equation  (6),  now  the  equation  of  the  tangent  of  the  sum  of  two  angles,  becomes 
cos  <?  [tan  (F+T)  +  tan  (F-T)] 


tan  2y     i+cos2  <p  tan  (F+D  tan  (F- i) 


(n) 


1  The  same  equation,  somewhat  differently  derived,  is  given  by  Rosenbusch-Wiilnng 
(Mikroskopische  Physiographic,  Ii,  4th  ed.,  1904,  253-254). 

G.   Cesaro  (Sur  une  methode  simple  pour  chercher  la  variation  ce  V angle  d' extinction  dans 
les  diffetentes  jaces  d'une  meme    zone.     Mem.    Acad.  Royale  de    Belgique,    3d    paper, 
LIV  (1896),    26  pp.     Read  July  7,  1894)  gives  an 
equation  expressed  in  terms  of  the  distance  OA  =  a 

(Fig.  495),  OB  =  b,OC  =  c,  and  the  angle  RCO  =  0.  /  n, '          T       .-^0 

A  and  B  are  the  points  of  emergence  of  the  optic 
axes,  IV  the  bisectrix  of  the  angle  in  the  new 
plane,  and  VIC  =  x. 

_  (a  +6)  sin/?+2ccosft 

This    equation   is    the    general    equation    for 
crystals  of  any  system.     If   the  axis  of   the  zone 
lies  in  the  plane  of  the  optic  axes,  c  =  o,  and 
(q+ft)  cos  (9o°-.5) 
i—ab  cos2  (90°—^) 
which  is  the  equation  for  the  100-010  zone. 

If  the  axis  of  the  zone  lies  in  either  of  the  other  two  principal  sections,  b=  —a,  and 

2C  COS  ft  fr.^ 

tan  2x= — ; — « — ~. — ;r~o •> 9~b  \**f 

i+a2  sin2  p— cz  cos2  p 

which  is  the  equation  for  the  001-010  zone. 

These  equations  are  extremely  simple  after  the  values  of  a  and  b  have  been  determined. 


tan  2x  = 


FIG.  495. 


402 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  349 


It  must  be  remembered  that  V  —  T  will  be  negative  in  this  case,  since  F>  V, 
therefore,  since  the  tangent  of  a  negative  angle  is  equal  to  minus  the  tangent  of  the 
same  positive  angle, 

tan  (V-T)=  -tan  (T-F), 


and  the  equation  may  be  written 


=  cos  y[tan  (F+T)- tan_(r -  F) ] 
27~i-cos2  <p  tan  (F+T)  tan  (T-F) 


(12) 


This  is  the  equation  for  extinction  angles  (y)  in  any  section  of  a  monoclinic  crystal 
whose  value  for  r<  F. 

Two  practical  applications  may  be-made  of  these  formulae:  (a)  The  extinc- 
tion angles  in  cleavage  flakes  of  minerals  of  a  group,  such  as  the  amphiboles 
or  pyroxenes,  may  be  computed;  (b)  The  extinction  angles  in  any  section 
of  a  zone  of  a  mineral  may  be  determined. 

As  an  example  of  the  first  we  may  take  the  pyroxenes,  whose  cleavage 
angle  (2^)  is  92°  54'.  In  the  following  table1  the  first  column  gives  values  of 
F,  the  extinction  angle  on  oio  for  the  particular  pyroxene  under  examination; 
the  other  columns  give  the  extinction  angles  on  no  for  different  values  of  the 
optic  angle. 

EXTINCTION  ANGLES  ON   no  CLEAVAGE  PLATES  OF  PYROXENES 


F  on  oio 

Values  of  2V  for 

50° 

60° 

70° 

35° 

29i 

30* 

31* 

36 

3oi 

3i 

32| 

37 

3if 

32- 

33f 

38 

3H 

33^ 

34f 

39 

33i 

34; 

3Sf 

40 

34i 

35; 

36f 

4i 

35 

36, 

37f 

42 

36 

37l 

38! 

43 

37 

38; 

* 

39t 

44 

38 

39:: 

4of 

45 

39 

40^ 

4if 

46 

40 

41? 

42f 

47 

4i 

42; 

44 

48 

42 

43  i 

45 

49 

43* 

44: 

46 

50 

44i 

45: 

47 

5i 

45i 

46^ 

48 

52 

46i 

47^ 

494 

53 
54 

47i 
48^ 

48^ 
49^ 

Soi 

sii 

Alfred  Harker:  Extinction  angles  in  cleavage-flakes.  Mineralog.  Mag.,  X  (1893), 
239-240. 

R.  A.  Daly:  On  the  optical  characters  of  the  vertical  zone  of  amphiboles  and  pyroxenes, 
etc.  Proc.  Amer.  Acad.,  XXXIV  (1899),  313- 


ART.  350] 


CALCULATION  OF  EXTIXCTIOX  ANCLES 


403 


As  an  example  of  the  second  application,  we  may  use  diopside  with  2V  — 
60°,  F  on  010  =  39°.  The  values  computed  from  formula  (12)  are  given  in 
the  following  table  and  are  graphically  shown  in  Fig.  496. 

EXTINCTION  ANGLES  IN  THE  100-010  ZONE 


• 

7 

0° 

39° 

o' 

10° 

38° 

47' 

20° 

38° 

06' 

30° 

36° 

57' 

4C° 

35° 

09' 

50° 

32° 

29' 

60° 

28° 

n' 

70° 

22° 

24' 

80° 

12° 

57' 

90° 

0° 

10 


A 


10  20  30  40  50  60  70  80  90 

PIG.  496. — Curve  showing  extinction  angles  in  the  100-010  zone  of  diopside. 


350.  Calculation  of  the  Extinction  Angle  for  any  Face  of  any  Zone  of  any 
Crystal. — This,  the  most  general  case  of  extinction  angles  in  zones,  was  first 
worked  out  by  Michel-Levy1  in  1877.  Other  formulae  are  given  by  Ferro2 

1  A.  Michel-Levy:  De  I'emploi  du  microscope  polarisant  a  lumiere  paroMele.     Ann.  d. 

Mines,  XII  (1877),  392-471. 

Review  by  H.  Bucking:  Ueber  die  Schwingungsricktungen   zweiaxiger   Kryslailplatlen 

und  dercn  AbMngigkeit  von  der  Richtung  der  Flatten.     Zeitschr.  f.  Kryst.,  Ill  (1878-9), 

217-231. 

A.  Michel-Levy  et  Alf.  Lacroix:  Les  mineraux  des  roches.     Paris,  1888,  9-40. 
-  A.  A.  Ferro:  Rivista  di  mineraligia,  etc.,  Padua,  XX  (1898),  i-n,  11-14.* 
Idem:  Atti  Soc.  Ligustica  di  Sc.  Xatur.,  Geneva,  IX  (1898),  143-       ,  230- 
Review  of  two  preceding  papers,  Zeitschr.  f.  Kryst.,  XXXII  (1899-1900),  532. 


404 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  350 


cot  27  = 


and  by  de  Sousa-BrandSo.1    That  of  Michel-Levy  is 

[cos  n  cos  v  —  sin  /*  sin  v  cos2  v]  +[sin  ^  sin  i>]  sin2  <p 
[cos  v  sin  GU+U)]  cos  ^>  —  [sin  v  sin  (ju  —  v)]  sin  ^> 
in  which  7  =  extinction  angle  in  the  section  under  examination, 

tp  =  the  angle  giving  the  position  of  this  section,  measured  from  a 
plane  passing  through  the  axis  of  the  zone  and  the  bisectrix 
of  the  acute  optic  angle, 

IJL  =  angle  between  the  axis  of  the  zone  and  one  optic  axis, 
v  =  angle  between  the  axis  of  the  zone  and  the  other  optic  axis, 
2V  =  angle  on  the  plane  under  examination  made  by  its  intersection 
with  the  two  planes  passing  through  the  optic  axes  and  the 
axis  of  the  zone. 


llx, 


FIG.  498. 
FIG.  497. 

Let  the  plane  of  the  paper,  Fig.  497,  represent  the  plane  of  the  optic  axes,  OA 
and  OB,  of  a  sphere  having  a  radius  of  unity.  Z  is  the  piercing  point  on  the  sphere 
of  the  axis  of  the  zone  in  which  the  extinction  angles  are  to  be  determined,  v  and 
IJL  are  the  distances  between  Zand  the  points  where  the  optic  axes  emerge  from  the 
sphere  (ZB  and  ZA).  Let  27^90°,  and  U+M^  180°.  Let  2V+n+v=-2p,  then  in 
the  spherical  triangle  ABZ 


cos  2V  =  cos  M  cos  u  +  sin  M  sin  'v  cos  2V, 


and 


tan  v  = 


sin 


—  ft)  sin  (p  —  u) 
sin  p  sin  (p—2V) 


If  the  axis  of  the  zone  lies  in  the  plane  of  the  optic  axes, 
y  =  o°  or  90°,  n=*=  v=  2V. 

1  Vicente  de  Sousa-BrandSo:  Sur  la  determination  de  V angle  des  axes  optiques  dans  les 
miner  aux  des  roches.  CommunicacSes  da  direccao  dos  Services  geologicos,  Lissabon,  IV 
(1900),  35-40. 

Idem:  Sur  la  determination  de  la  position  des  axes  optiques  au  may  en  des  directions  d' ex- 
tinction. Ibidem,  41-56. 


ART.  350]  CALCULATION  OF  EXTINCTION  ANGLES  405 

When  the  axis  of  the  zone  lies  in  the  plane  of  the  bisectrix  b  and  the  axis  of  least 
ease  of  vibration  «, 

/z=i8o°—  u     or  /z-f  v=  i8c.° 

Let  the  angle  between  that  section  of  the  zone  in  which  the  extinction  is  to  be 
determined  (QR,  Fig.  498)  and  the  plane  of  the  bisectrix  (ZP)  be  y>.  Take  as  the 
plane  of  the  drawing  the  plane  at  right  angles  to  the  axis  of  the  zone  Z.  It  contains 
the  normal  N.  Draw  circles  (appearing  as  straight  lines  in  the  projection  of  the 
figure)  through  ZB  and  ZA,  cutting  the  circumference  of  the  sphere  at  b'  and  a'. 
Draw  great  circles  through  NB  and  NA,  cutting  QR  at  b  and  a,  and  draw  a  plane 
bisecting  BNA.  The  trace  of  this  bisecting  plane  (Nc)  cuts  QR  at  c,  and  its  in- 
tersection with  the  QR  plane  is  the  direction  of  extinction.1  If  the  angle  cZ  =  y, 
then  from  the  figure, 

aZ+bZ 

cZ  =  y  =  -—-f 

But  aZ  =  ZNA=go°-ANa', 

and  in  the  spherical  right  triangle  AN  a' 


sm  Na      cos 

from  which 

tan  aZ  =  tan  M  cos  (?>+»). 
In  the  same  manner 

tan  &Z  =  tan  v  cos  (<?  —  ?), 
whereby,  finally 

f  (*7J-^7\      i  "tan  //  tan  u  cos  (?+»)  cos  (y-tQ 

COt   2-y  =  COL   (0Z  +  0Z)  =  —  7  -  r-  ,  (2) 

tan  /<  cos  (?+»)  +  tan  u  cos  (p—  ») 

or,  developing  the  value  of  the  tangents  into  cosine  values, 

• 

(cos  fi  cos  y  —  sin  p.  sin  u  cos2  v)  -f-  (sin  /*  sin  a)  sin2  <p 
[cos  v  sin  (jP-f-cj)]  cos  ^  —  [sin  v  sin  (p  —  u)]  sin  ^ 

The  value  of  v  may  be  determined  from  equation  (i)  and  inserted  in  (3).  If  we 
substitute  the  following  letters  for  the  values  that  remain  constant  in  the  same 
zone, 

A  =cos  n  cos  v  —  sin  n  sin  u  cos2:, 

E=  sin  M  sin  v  ,- 

C=  cos  v  sin  (/*+u), 

Z)=  sin  »  sin  (/*—  u). 


sin2  ? 

the  equation  becomes  cot  2y—   „  —  _>    .  U) 

C  cos  ?  —  D  sm  <r 

From  this  equation  may  be  calculated  the  extinction  angle  for  any  section  in  the 
same  zone  if  the  angles  n,  y,  and  <p  are  known. 

i  Cf.  Art.  349- 


406 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  351 


PIG.  499. 


351.  Graphical  Methods  for  the  Determination  of  Extinction  Angles 
on  any  Plane. — The  extinction  angles  on  different  faces  of  a  zone  may  be  de- 
termined much  more  easily  by  means  of  stereographic  projection  than  by 
analytical  methods,  as  was  well  shown  by  Michel-Levy1  in  1894.  The 
method  has  been  used  by  many  writers  since  then,  among  others  by  von 
Fedorow,  Wulff,  Viola,  and  especially  by  Duparc  and  Pearce,2  who  give  the 
following.  , 

We  know,  from  the  law  first  determined  experimentally  by  Biot3  and 
proved  theoretically  by  Fresnel,4  that  the  direction  of  extinction  on  a 

section  is  that  in  which  the  section  is  cut  by 
the  trace  of  the  plane  bisecting  the  angle  be- 
tween the  planes  through  the  optic  axes  and 
the  normal  to  the  section.  If,  then,  A  and  B, 
Fig.  499,  are  the  poles  of  the  two  optic  axes, 
and  P  is  that  of  any  section  whose  trace  is 
Dacb,  the  traces  PA  and  PB  will  represent 
the  planes  through  the  optic  axes,  and  PC  and 
Pd,  bisecting  bPa  and  bPaf,  the  bisecting 
planes  whose  intersections  with  the  plane  acbd 
are  the  lines  of  extinction.  In  the  stereo- 
graphic  projection,  c  and  d  will  be  the  poles 
of  these  extinctions  and  the  arcs  from  c  and  d  to  x,  the  point  of  emer- 
gence of  some  crystallographic  line  (Ox,  Fig.  449),  will  represent  the  ex- 
tinction angle  on  that  plane. 

The  construction  is  very  simple  if  the  axis  of  the  zone  lies  at  right  angles 
to  the  plane  of  the  drawing,  Fig.  500.  In  this  case  the  great  circles  through 
the  axis  are  represented  by  straight  lines,  since  all  the  planes  of  that  zone  are 
perpendicular  to  the  plane  of  projection.  The  axis  of  the  zone  is  in  the 
center.  The  method  for  changing  the  plane  of  the  projection  is  given  in 
Article  16,  Problem  9. 

The  extinction  angle  on  any  plane,  such  as  SS'  (Fig.  500)  of  the  zone  of 
which  Z  is  the  axis,  may  be  determined,  according  to  Fresnel's  law,5  by 
passing  through  the  normal  to  that  plane  (NN'),  two  planes,  each  containing 

1  A.  Michel-Levy:  Etude  sur  la  determination  des  j 'elds paths.     I,  Paris,  1894,  16-18. 

2  F.  Duparc  und  F.  Pearce:  Ueber  die  Ausloschungswinkel  der  Flachen  einer  Zone. 
•Zeitschr.  f.  Kryst.,  XLII  (1905-7),  34-46. 

Idem:  Traite  de  technique  miner  alogique  et  petrographique.    Leipzig,  1907,  249-260. 

3  J.  B.  Biot:  Memoir e  sur  les  lois  generates  de  la  double  refraction  et  de  la  polarisation, 
dans  les  corps  regulierement  cristallisees.    Mem.  Acad.  France,  Ann6e  1818,  III  (1820), 
177-384. 

4  A.  Fresnel:    Memoire  sur  la  double  refraction.    Ibidem,  VII  (1827),  45-176. 

Idem:  Ueber  die  doppelte  Strahlenbrechung.    Translation  of  preceding.     Pogg.    Ann., 
XXIII  (1831),  372-434,  494-560.     In  particular  542-545. 
6  Cf.  Arts.  349  and  350. 


ART.  351] 


CALCULATION  OF  EXTINCTION  ANGLES 


407 


PE' 


the  normal — consequently  lying  at  right  angle  to  the  plane  SS' — and  one  of 
the  optic  axes  (A  or  B).  If,  now,  a  plane  is  passed  through  N,  bisecting  the 
angle  between  the  two  planes  containing  the  optic  axes,  its  trace  on  SS'  will 
represent  the  extinction  angle  on  that  plane.  In  the  stereographic  projec- 
tion these  planes  are  represented  by  great  circles  passing  through  NN'  and 
A  or  B. 

To  determine  the  position  of  the  bisecting  plane,  two  methods  may  be 
used.  The  projection  of  the  optic  angle  on  the  plane  SS'  is  measured  by 
the  arc  LR,  and  may  be  read  directly  if  a  stereographic  net  is  used.  Half 
the  angle  (LE  =  ER)  gives  the  point  E,  which  represents  the  position  of  the 
line  of  extinction.  The  angle  ZE  is  the  required  angle  of  extinction.  If 
no  stereographic  net  is  used,  the  value 
may  be  obtained  by  drawing  lines  through 
N  and  the  points  L  and  R  where  the 
great  circles  N'AN  and  N'BN  cut  SS' 
and  continuing  them  to  the  circumference 
of  the  circle  at  L'  and  R'.  The  arc  L'R' 
measures  the  projection  (  =  2v)  of  the 
optic  axial  angle  (  =  27)  on  SS',  and  its 
bisectrix  NE'  gives  the  angle  of  extinc- 
tion, measured  by  the  arc  N'E'  or  ZE. 
The  projection  of  the  other  extinction 
direction  likewise  lies  on  SS'  and  at  a 
distance  of  90°  from  the  first  (E'Fr  =  go°). 
Its  projection  is  at  F. 

The  extinction  angles  of  all  possible  planes  in  the  zone  about  Z  can  be 
determined  by  rotating  the  plane  SS'  about  the  axis  Z,  whereby  a  series  of 
points,  analogous  to  E  and  F,  representing  the  points  of  emergence  of  the 
extinction  line  for  all  planes  in  the  zone,  will  be  obtained  (Fig.  501).  These 
points  will  lie  on  two  curves,  and  will  be  so  placed  that  on  every  plane  through 
the  zonal  axis  they  will  be  separated  by  90°.  The  lengths  of  the  arcs  con- 
necting any  point  with  the  point  Z  represents  the  extinction  angle  in  the 
plane  whose  trace  coincides  with  the  arc.  In  one  direction  the  extinction 
will  be  toward  the  slowest  ray  and  in  the  other  toward  the  fastest. 

Several  cases  may  occur,  depending  upon  the  position  of  the  axis  of 
the  zone.1 

I.  If  the  axis  of  the  zone  occupies  a  random  position  in  relation  to  the 
position  of  the  optic  axis,  the  extinction  curve,  for  a  value  of  2F  =  6o°,  is  as 
shown  in  Fig.  501.  This  may  be  developed  on  rectangular  coordinates  as 
shown  in  Fig.  502,  in  which,  as  in  the  preceding  figure,  the  solid  line  indicates 
extinction  angles  from  the  zonal  axis  to  the  acute  bisectrix  (Bxa)  and  the 


1  See  Duparc  and  Pearce,  Op.  cit.,  for  a  mathematical  discussion  of  the  various  curves 
produced. 


408 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  351 


dotted  line  to  the  obtuse  (Bx0),  the  plane  SSf  being  taken  as  the  initial 
point. 

II.  If  the  axis  of  the  zone  lies  in  the  plane  which  passes  through  the  acute 


40 

20 
10' 

0 

-10 
--'0 

-40 
-50 
-bO 
-70 

-ao 
-yo 

-( 

/* 

9 

~^ 

'  —  . 

•N 

^ 

/ 

h^ 

^ 

^ 

/ 

*s 

[/ 

s 

fc 

/ 

\ 



S 

' 

*"• 

%^ 

x 

"V 

""s. 

> 

^ 

x. 

' 

v 

/ 

\ 

ou 

1  10  "20°  30"40050060070080°90'lOO'lJ 

01: 

0130  iJ 

0"J6017018( 

PIG.    501. — Extinction    curve    for     zV  = 
60°,  axis  of  zone  in  random  position. 


PIG.  502. — Extinction  angles,  derived  from  the 
stereographic  projection  of  preceding  figure,  developed 
on  rectangular  coordinates. 


bisectrix  and  the  direction  of  mean  ease  of  vibration  (b),  the  extinction  angles 
are  as  shown  in  Fig.  503.     The  position  of  the  axis  of  mean  ease  of  vibration 


TO" 

80° 
50° 
40^ 
30° 
20 

ioq 

0° 

-ioc 

-20C 
-30C 
-40' 
-50C 
-60 
-70' 

-so' 

-  DO 

/ 

^" 

~-v 

x 

/ 

N 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

\ 

,' 

•»"" 

'-« 

•% 

/ 

\ 

/ 

\ 

/ 

\ 

/ 

s 

s 

xx 

0-C 

"  10"  20  30  40  50  80  70"  80°  90  lOOllti  I2ff\30  "iW  ISO  180  170  18 

FIG.  503. — Extinction  curve  when  the  axis  FIG.    504. — Extinction  angles,  derived  from  the 

of    the    zone    lies  in   the  plane  which  passes  stereographic  projection  of  preceding  figure,  developed 

through  the  acute    bisectrix  and  the  direction  on  rectangular  coordinates, 
of  mean  ease  of  vibration. 

is  determined  by  the  point  of  intersection  of  the  two  planes  whose  poles  are 
the  two  bisectrices  (Bxa  and  Bx0)  of   the  optic  axial  angles.     Developing 


ART.  351] 


CALCULATION  OF  EXTINCTION  ANGLES 


409 


the  curve  on  rectangular  coordinates,  and  using  the  trace  of  the  Bxa-Bx0 
plane  as  the  initial  line,  we  have  the  curve  shown  in  Fig.  504. 

III.  The  axis  of  the  zone  lies  in  the  plane  passing  through  the  axis  of  inter- 


=0  10°20  30°  40" 50°  60"  70°  80 '  90  100 110 120 130 140 150  160170  180 


FIG.  505. — Extinction    curve   when  the  FIG.    506. — Extinction     angles,    derived    from    the 

axis  of  the  zone  lies    in  the  plane  passing  stereographic  projection  of  preceding  figure,  developed 

through  the  intermediate   ease  of  vibration  on  rectangular  coordinates, 
and  the  obtuse  bisectrix. 


mediate  ease  of  vibration  and  the  obtuse  bisectrix, 
are  as  shown  in  Figs.  505  and  506. 


The  extinction  angles 


FIG.  507.  FIG.  508. 

FIGS.  507  and  508. — Extinction  curve  when  the  axis  of  the  zone  lies  in  the  plane  of  the  optic  axis. 
Fig.  507.  The  zonal  axis  falls  in  the  quadrant  containing  the  acute  bisectrix.  Fig.  508.  The  zonal 
axis  falls  in  the  quadrant  containing  the  obtuse  bisectrix. 

IV.  The  axis  of  the  zone  lies  in  the  plane  of  the  optic  axes.  There  are 
two  cases,  (a)  the  zonal  axis  falls  in  the  quadrant  containing  the  acute 
bisectrix  (Fig.  507),  and  (b)  it  falls  in  the  quadrant  of  the  obtuse  bisectrix 
(Fig.  508). 


410 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  352 


PROBLEMS 

Construct,  on  rectangular  coordinates,  the  extinction  curves  shown  in  Figs. 
507  and  508. 

Construct,  first  in  stereographic  projection,  then  on  rectangular  coordinates,  the 
extinction  curve  for  the  100-001  zone  of  diopside. 


FIG.  509. — Extinction  angles  in  andesine  (AbssAna?),  shown  in  stereographic  projection  at  the  poles 

.      of  the  faces. 

352.  Extinction  Diagram  and  Curves  of  Equal  Extinction. — Instead  of 
making  a  diagram  showing  the  different  extinction  angles  in  a  zone  by  means 
of  the  piercing  points  of  the  extinction  lines,  we  can  make  a  diagram  which 
gives  all  possible  extinctions  in  a  crystal.  These  extinction  angles  may  be 
shown,  in  stereographic  projection,  by  indicating  their  values  at  the  poles 
of  the  different  planes,  usually  at  the  intersection  of  every  tenth  parallel 
and  meridian  (Fig.  509). l  The  values  for  the  extinction  angles  in  the 
i oo-o  10  zone  will  thus  be  given  around  the  periphery  of  the  projection 
circle,  and  will  correspond  in  value  to  the  angles  shown,  by  the  previous 

1  After  Rosenbusch:  Mikroskopische  Physiographic,  4  Aufl.,  1905,  12,  plate  XVII. 


ART. 


CALCULATION  OF  EXTINCTION  ANGLES 


411 


construction,  in  Fig.  510.  The  extinction  angles  in  the  100-001  zone  are 
shown  along  the  vertical  diameter  (Fig.  509),  and  those  in  the  OIO-QOI  zone 
along  the  horizontal  diameter.  By  connecting  equal  values,  we  obtain 
curves  of  equal  extinction.  The  lines,  in  other  words,  represent  the  emer- 
gence of  the  poles  of  all  the  planes  in  which  the  extinction  angles  are  equal 


FlG.  510. — Construction  for  determining  the  extinction  angles  in  andesine. 

(Fig.  511).  Practical  use  is  made  of  these  curves  in  the  study  of  certain 
minerals,  notably  the  plagioclase  feldspars.1  They  are  also  used  in  the  von 
Fedorow2  method  for  determining  the  optic  axial  angle. 

PROBLEMS 

From  the  diagram  of  extinction  angles,  Fig.  509,  draw,  in  rectangular  co- 
ordinates, the  extinction  angles  in  the  100-001  zone. 

Make  a  diagram  of  equal  extinction  angles  for  diopside. 

1  Michel-Levy:    Etude  sur  la  determination  desjeldspaths  dans  les  plaques  minces.     Paris, 
1894,  I,  planches  I-V11. 

2  See  Arts.  427  et  seq. 


412 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  353 


353.  Influence  of  Dispersion  upon  Extinction  Angles. — The  property 
of  dispersion,  possessed  to  a  greater  or  less  degree  by  all  crystals,  has  its 


FIG.  511. — Curves  of  equal  extinction  in  andesine.  (Abea  Ansr). 

influence  upon  the  angles  of  extinction.     The  optical  ellipsoid  is  slightly 
differently   oriented   for   different   colors  (Fig.  512),  therefore  its  axes  will 
lie  in  different  positions  and,  consequently,  will  show  slightly 
different  extinction  angles  for  different  colored  light.     As  a 
result,  when  white  light  is  used,  there  will  be  no  position  of 
total  darkness  in  certain  crystals  which   possess  high   dis- 
persion,  since  the  rays  do  not  all  extinguish  together.     If 
monochromatic  light  be  used,  the  extinction  angles  will  be 
slightly  different  for  different  colors. 
FIG.  512.— Dis-         This  dispersion  of  the  extinction  lines  is  called  dispersion 
persion  of  the  bi-  Oj  tfo  bisectrices,  since  the  extinction  lines  coincide  with  the 

sectrices  in  a  mono-  r     .. 

clinic  crystal.          bisectrices  of  the  optic  angles. 


CHAPTER  XXIX 
OBSERVATIONS  BY  CONVERGENT  LIGHT 

354.  Polariscope,  Conoscope. — Another  series  of  tests  may  be  made 
upon  minerals  by  observing  the  phenomena  produced  in  them  by  means  of 
convergent  polarized  light,  by  whose  interference,  under  certain  conditions, 
there  will  be  produced  a  figure.  Instruments  fitted  for  such  observations, 
and  consisting  of  polarizer  and  analyzer,  and  strongly  converging  lenses 
above  and  below  the  object  stage,  are  called  polariscopes1  or  conoscopes.2 
Usually  the  magnifying  power  of  such  instruments  is  not  great,  and  they  are 
used  for  observations  on  large  mineral  slices.  Being  rarely  used  for  making 
observations  in  petrographic  determinations,  they  will  not  be  described 
here.  The  petrographic  microscope,  however,  may  be  converted  into  a 
conoscope  by  using  a  medium  or  high-power  objective  and  inserting,  below 
the  stage,  a  converging-lens  system  (Figs.  255-260).  Such  lenses  were 
originally  inserted  in  metal  caps  which  were  placed  over  the  upper  end  of  the 
polarizer.  This  necessitated  the  removal  of  the  thin  section  or  the  with- 
drawal and  replacement  of  the  polarizer,  an  awkward  proceeding  with  some 
microscopes.  At  the  present  time  most  makers  insert  the  condensing  system 
on  pivots  or  sliders,3  the  object  being  to  be  able  to  change  rapidly  from 
parallel  to  convergent  light.4  Czapski5  suggested  that  it  is  possible  to  change 
from  parallel  to  convergent  light  by  simply  stopping  down,  by  means  of  a 
diaphragm,  the  light  coming  from  below.  He  says  that  although  it  reduces 
the  amount  of  light,  it  is  possible,  by  this  means,  to  obtain  as  good  interfer- 
ence figures  as  when  the  condensing  lenses  are  inserted.  An  objection  to  this 
method  is  that  the  size  of  the  field  is  greatly  reduced. 

The  passage  of  the  light  through  a  microscope  arranged  as  a  conoscope 
is  shown  in  Fig.  513.  The  light,  reflected  from  the  mirror,  is  plane  polarized 
on  passing  through  the  lower  nicol.  It  converges  in  a  cone  of  wide  angle 

1  G.  Kirchhoff:  Ueber  den  Winkel  der  opiischen  Axen  des  Aragonits  Jur  die  verschiedenen 
Fraunhofer'schen  Linien.     Pogg.  Ann.,  CVIII  (1859),  567-575. 

P.  Groth:  Ueber  Apparate  und  Beobachtungsmethoden  jur  krystallographisch-optische 
I'ntersuchungen.  Pogg.  Ann.,  CXLIV  (1871),  34-55. 

2  Gustav  Tschermak.  * 

3  Art.  1 1 8,  supra. 

4  H.  Laspeyres:  Vorrichtung  am  Mihoskope  zur  raschen  Umwandlung  paralleler  Licht- 
strahlen  in  convergente.     Zeitschr.  f.  Kryst.,  XXI  (1902),  256-257. 

5  S.   Czapski:  Ueber  Einrichtungen  behujs  schnellen    Ueber ganges   vom   parallelen   zum 
confer genten  Lichte  und  die  Beobachtung  der  Axenbilder  von  sehi  kleinen  Krystallen  in  Polari- 
sations-Mikroskopen.     Zeitschr.  f.  Kryst.,  XXII  (1893-94),  158-162. 

413 


414 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  354 


from  below  the  object,  passes  through  the  objective  and  the  analyzer,  and 
forms  a  real  image  #3  in  the  tube,  a  short  distance  above  the  analyzer.  As 
ordinarily  arranged,  this  image  cannot  be  seen  through  the  ocular,  since  it 
does  not  lie  in  its  focal  plane.  It  may  be  seen,  however,  if  there  is  inserted 


FIG.  513. — Passage  of  light  through  a  microscope  arranged  as  a  conoscope.     (Leitz.) 

an  accessory  lens,  as  shown  in  the  figure,  making,  with  the  ocular  above  it, 
a  weak  compound  microscope  in  itself.  The  image  may  be  seen,  likewise, 
by  removing  the  ocular  entirely  and  looking  down  the  tube,  or  it  may  be 
observed  by  placing  a  hand  lens  at  the  proper  distance  above  the  second 
image  which  is  formed  in  the  Ramsden  disk  above  the  ocular.  The  various 
methods  will  be  described  in  detail  below.1 
1  Arts.  389-401. 


ART.  356]  OBSERVATIONS  BY  CONVERGENT  LIGHT  415 

355.  Interference  Figures. — The  kind  of  image  formed  by  the  conoscope 
depends  upon  the  crystal  system  of  the  mineral  under  examination  and  upon 
the  orientation  of  the  section.     It  consists  of  curves  and  bars  (Figs.  520,  522, 
528,  etc.),  either  black,  or  black  and  colored,  depending  upon  whether  mono 
chromatic  or  white  light  is  used.     By  means  of  these  images,  called  inter- 
ference figures  since  the  bars  and  colors  are  produced  by  the  interference 
of  the  rays  which  have  traversed  the  crystal  in  different  directions,  it  is 
possible  to  separate  uniaxial  from  biaxial  crystals,  to  determine  the  direction 
of  the  optic  axes,  the  angle  between  them  in  biaxial  crystals,  the  directions 
of  the  fastest  and  the  slowest  rays  in  the  crystal,  the  character  of  the  disper- 
sion, and  the  orientation  of  the  section.     It  is  also  possible  to  determine 
by  them  the  amount  of  the  retardation,  consequently,  if  its  thickness  is 
known,  the  value  of  the  birefringence  of  the  mineral. 

By  convergent  light  we  may  divide  all  crystals  into  three  groups,  two  of 
of  which  may  again  be  subdivided: 
Isotropic  crystals. 

Uniaxial  crystals — positive — negative. 
B  iaxial  crystals — positive — negative . 

ISOTROPIC  CRYSTALS 

356.  Random  Sections. — We  saw  that  in  parallel  polarized  light,  be- 
tween crossed  nicols,  an  isotropic  crystal  remained  dark  during  a  complete 
rotation  of  the  stage.     Upon  altering  the  light  from  parallel  to  convergent, 
no  change  takes  place  in  the  appearance  of  the  field  of  view.     The  light, 
passing  through  with  equal  ease  in  every  direction,  has  no  effect  upon  the 
plane  of  polarization  of  the  light  entering  from  below,  consequently  it  is 
completely  cut  off  by  the  analyzer  and  darkness  results.     The  mineral 
under  examination  must  be  either  amorphous  or  belong  to  the  isometric 
system. 

In  practice  the  light  is  never  completely  polarized,  for  a  beam  of  plane  polarized 
light,  falling  at  a  considerable  inclination  upon  an  isotropic  substance,  such  as  glass, 
suffers  a  certain  amount  of  polarization,  and,  as  a  consequence, 
the  emerging  light  no  longer  vibrates  in  a  single  plane.  The 
greater  the  inclination  of  the  rays,  the  greater  the  polarization  of 
the  emerging  light,  whereby,  as  was  shown  by  Rinne,1  the  light  is 
polarized  in  directions  at  right  angles  to  the  radii  at  the  edges  of 
lenses,  especially  those  of  short  focal  lengths.  The  planes  of  FIG.  514— Polar- 
polarization  of  these  rays  are  thus  represented  by  the  radiating  izati°n  of  light  by 

'  lenses. 

lines  in  Fig.  514. 

1  F.  Rinne:  Bemerkung  iiber  die  Polarisationswirkung  von  Linsenrdndern.  Centralbl. 
f.  Min.,  etc.,  1900,  88-89. 

See  also  G.  Cesaro:  Etude  de  la  rotation  imprimee  an  plan  de  polarisation  du  faisceau 
lumineiix  vcnant  du  polariseur,  par  les  lentilles  du  microscope  a  lumicre  con-oergente.  Bull. 
Acad.  Roy.  Belgique  Cl.  d.  Sci.,  1906,  459-492. 


416 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  357 


The  effect  of  this  polarization  by  the  lenses  may  be  seen  if  an  isotropic  mineral 
or  glass  is  examined  for  its  interference  figure.  With  most  microscopes  there  will 
appear  a  broad,  apparently  uniaxial,  cross,  usually  of  weak  positive  character. 
The  same  cross  will  appear  if  no  mineral  lies  upon  the  stage,  wherefore  care  must 
be  taken  in  regard  to  this  figure  so  that  it  may  cause  no  confusion.  If  the  inter- 
ference figure  of  a  biaxial  crystal,  having  a  large  optic  axial  angle,  such  as  muscovite, 
adularia,  etc.,  and  cut  at  right  angles  to  the  acute  bisectrix,  be  examined,  it  will 
be  found  that  if  the  upper  nicol  is  removed,  the  interference  figure  will  still  be  seen 
around  the  edges  of  the  field,  though  somewhat  dimmer  than  before. 

ANISOTROPIC  CRYSTALS 
UNIAXIAL  CRYSTALS 

357.  Section  Perpendicular  to  the  Optic  Axis. —  Let  us  consider,  first,  a 
basal  section  of  a  uniaxial  crystal.  From  the  optical  ellipsoid  we  know  that 
such  a  section  lies  at  right  angles  to  the  optic  axis,  and  that  the  rays  passing 
through  it  vibrate  with  the  same  ease  in  every  direction.  Rays  passing 


e'd  c'   b     a        0        a      b    c  <J  e 


J/  Al 

FIG.  515.  FIG.  516.  FIG.  517. 

FIGS.  515  to  517. — Interference  of  convergent,  polarized  light  in  uniaxial  crystals.      (After  Miers.) 

through  it  in  any  other  direction  are  doubly  refracted  and,  consequently, 
interfere.  Suppose  a  cone  of  monochromatic  light  passes  through  a  crystal 
section  -(Fig.  515)  cut  at  right  angles  to  the  optic  axis  MO.  The  ray  MO 
has  its  vibrations  equal  in  all  directions  and  perpendicular  to  this  axis.  It 
passes  through  without  interference.  At  this  point  of  emergence  (O,  Fig.  517) 
the  section  will  appear  dark.  The  ray  Ma,  passing  through  the  crystal  at  an 
angle  with  the  optic  axis,  is  doubly  refracted.  Let  the  inclination  of  this 
ray  be  such  that,  owing  to  its  greater  path  difference  and  to  the  greater 
difference  of  refractive  indices  of  the  two  rays  in  this  direction,  the  retarda- 
tion is  exactly  one  wave  length.  We  saw1  that  rays  with  a  retardation  of  N\ 
were  extinguished  between  crossed  nicols,  consequently  at  the  points  a  and 
a',  where  the  retardation  is  one  wave  length,  there  will  be  darkness.  In  a 

1  Art.  282,  supra. 


FIG.  522.  PIG.  523. 

FIG.  518. — Dependence  of_the  size  of  an  interference  figure  upon  the  numerical  aperture  of  the 
condenser.  Topaz,  section  at  right  angles  to  the  acute  bisectrix,  placed  in  diagonal  position  between 
crossed  nic;ls.  X.  A.  of  condenser  0.636. 

FIG]  519-—  Ditto.     N.  A.  1.168. 

FIG.  520. — Dependence  of  the  curves  of  equal  retardation  upon  the  color  of  light  used.  Calcite, 
1/4  mm.  thick,  cut  at  right  angles  to  the  optic  axis.  Between  crossed  nicols.  Wave  length  of  source 
of  light,  620-720/1/1  (blue). 

FIG.  521. — Ditto.     Wave  length  of  source  of  light,  410-450/1/1  (yellowish  orange). 

FIG.  522. — Ditto.     Cerussite,  biaxial.     Wave  length  of  source  of  light,  620-720/1/1  (blue). 

FIG.  523. — Ditto.     Wave  length  of  source  of  light,  410-450/1/1  (yellowish  orange). 

(Facing  Page  416.) 


ART.  357]  OBSERVATIONS  BY  CONVERGENT  LIGHT  417 

cone  of  light,  such  as  we  are  considering,  there  will  be  an  equal  retardation 
of  one  wave  length  everywhere  at  a  distance  of  Oa  from  the  optic  axis, 
whereby  a  circle  of  darkness,  aa',  Figs.  516-517,  will  appear  at  that  distance. 
Another  ray  Ob  has  a  retardation  of  2\,  consequently  b  and  bf,  Fig.  515,  or 
the  circle  bbrt  Figs.  516-517,  will  appear  dark.  Other  rays  will  interfere 
with  phasal  differences  of  3\,  4\,  5\,  etc.,  forming,  thus,  concentric  rings 
between  which  there  will  be  rings  of  light. 

If  a  different  color  of  light  from  that  used  in  obtaining  the  above  rings 
were  used,  we  should  find,  owing  to  the  difference  in  their  wave  lengths,  that 
the  distances  Oa,  Ob,  Oc,  etc.,  would  be  different,  consequently  the  rings 
would  be  farther  apart  with  blue  light  (Figs.  520  and  522),  and  closer  together 
with  yellow  (Figs.  521  and  523). 

The  number  of  rings  which  may  be  seen  in  the  field  of  the  microscope 
likewise  depends  upon  the  value  of  the  double  refraction  of  the  mineral 
(Figs.  590-5-91),  the  numerical  aperture  of  the  condenser  (Figs.  518-519), 
and  the  thickness  of  the  section  (Figs.  591-592).  Thus  a  section  of  quartz 
i.o  mm.  in  thickness  (Fig.  590)  will  show  two  isochromatic  curves,  while  a 
section  of  calcite  of  the  same  thickness  (Fig.  591)  will  show  a  great  many. 
If  the  quartz  section  is  increased  in  thickness,  a  point  will  be  reached  where 
the  same  number  of  rings  is  seen.  Knowing  the  thickness  of  the  section, 
it  is,  consequently,  possible  to  determine  approximately  the  maximum  bire- 
fringence of  a  mineral  from  a  section  wrhich,  perhaps,  show  no  double  refrac- 
tion at  all. 

If  white  light  were  used  instead  of  monochromatic,  bands  of  color  would 
appear  everywhere  instead  of  darkness  and  light,  each  color  representing 
the  complementary  one  of  that  which  was  extinguished  at  that  point,  the 
color  being  the  same  as  that  which  would  be  produced  by  the  retardation 
in  a  section  of  the  same  thickness  cut  at  right  angles  to  the  direction  of  the 
ray.  Thus  the  color  at  a  (Fig.  515)  will  be  the  same  as  that  which  would  </ 
be  produced  by  a  plate  of  a  thickness  Ma,  cut  at  an  angle  OM a  with  the  optic 
axis,  and  viewed  between  crossed  nicols  in  parallel  polarized  light. 

But  besides  these  dark  (or  colored)  rings,  there  appear,  also,  in  the  inter- 
ference figures  of  uniaxial  crystals  (Figs.  520-521),  certain  dark  bars,1  due 
to  the  relation  of  the  differently  orientated  vibration  directions  to  the  princi- 
pal planes  of  the  nicols.  These  curves  of  like  vibration  directions  are  called 
isogyres.  They  may  be  explained  graphically  as  follows: 

Let  a  certain  amount  of  light,  represented  by  the  horizontal  line2  ab,  Fig.   & 
524,  enter  the  crystal  plate.     Since  the  ray  is  now  passing  through  at  an  angle 
with  the  optic  axis,  the  ease  of  vibration  is  not  the  same  in  every  direction, 
but  the  light  vibrates  in  planes  at  right  angles  to  each  other.     Let  these  two 

1  Cf.  Art.  371. 

2  The  line   actually  represents  the  square  root  of  the  intensity,  as  we  shall  see  later. 
This  applies  to  Fig.  525  also. 

27 


418 


MANUAL  OF  PETROGRAPHIC  METHODS 


"[ART.  358 


directions  be  ac  and  ad.  Resolving  the  original  intensity  of  the  light  into 
these  two  directions,  we  have  ac  and  ad  as  the  values.  Leaving  the  crystal, 
the  light  passes  to  the  upper  nicol,  which  has  its  vibration  direction  at  right 
angles  to  the  polarizer.  The  rays  ac  and  ad  are  each  resolved  into  two  rays, 
parallel  and  at  right  angles  to  the  analyzer;  the  former  components,  ae  and 
af,  pass  through,  but  the  latter  are  annihilated.  There  reach  the  eye,  there- 
fore, only  the  rays  ae  and  af,  and  their  sum  represents  the  intensity  of  the 
light  reaching  the  eye  through  the  crystal  at  that  point.  Applying  a  similar 
construction  to  various  rays  in  Fig.  525,  we  see  that  all 
points  on  any  radial  line  from  O  have  the  same  intensity. 
This  intensity  is  at  its  maximum  on  the  line  making  an 
angle  of  45°  with  the  vibration  directions  of  the  two  nicols, 
and  is  equal  to  zero  when  parallel  to  them.  As  a  result, 
there  will  be  a  gradation  of  light  from  the  maximum  inten- 
sity in  the  diagonal  positions,  to  darkness  when  the  vibra- 
tion planes  are  parallel  to  the  principal  sections 
of  the  nicols. 

If  the  nicols  are  placed  in  parallel  position, 
the  maximum  amount  of  light  will  pass  along 
the  vibration  planes,  and  instead  of  a  dark  cross 
(Fig.  526),  there  will  be  one  of  light  (Fig.  527). 
The  greatest  darkness  in  the  interference  figure 
will  be  on  the  diagonals. 


FIG.  524. 


FIG.  525. — Intensity  of  emerg- 
ing light  and  the  cause  of  the  ap- 
pearance of  the  isogyres  in  a  uni- 
axial  interference  figure. 


Analytically  we  have,  for  the  equation  of  the  in- 
tensity of  light  at  any  point  in  a  crystal  plate  be- 
tween crossed  nicols, 


sin2  26  sin2 

A 


(Eq.  16,  Art.  285), 


where  r  is  the  amplitude  of  vibration,  6  the  angle  between  the  vibration  planes  of 
the  mineral  and  the  polarizer,  and  M(n-i  —  ni)  the  retardation.  The  amplitude, 
the  wave  length,  and  the  retardation  remain  the  same  for  any  circle  around  the 
point  0,  whereby  the  equation  may  be  written 

I  =  K  sin2  26. 

The  intensity,  therefore,  depends  directly  upon  the  value  of  sin2  26.  The  maximum 
value  for  the  sine  of  an  angle  is  unity,  consequently  the  maximum  intensity  of  the 
light  is  where  sin2  26=  i.  From  this  equation  we  obtain  20  =  90°,  or  0  =  45°.  The 
minimum  intensity  occurs  where  sin2  26  =  0  which  occurs  when  20  =  oor  180°,  and 
6  =  o  or  90°.  These  results  are  the  same  as  those  derived  from  the  graphical  method 
above. 

358.  Section  Oblique  to  the  Optic  Axis. — If  the  axis  of  the  crystal  is 
perpendicular  to  the  section,  the  center  of  the  cross  is  on  the  axis  of  the 
microscope  (Fig.  526),  and  no  matter  whether  the  slide  is  displaced  laterally 


Fro.  530.  FIG.  531. 

Fir,.  526. — Interference  figure  of  calcite.  Plate  1/2  mm.  thick,  cut  at  right  angles  to  the  optic 
axis.  In  sodium  light  with  nicols  crossed. 

FIG.   527. — Ditto.     In  white  light  with  nicols  parallel. 

FIG.  528. — Calcite  plate,  section  cut  at  an  angle  of  80°  with  the  optic  axis.  Nicols  crossed,  optic 
axis  of  crystal  lying  in  the  vibration  plane  of  the  analyzer;  sodium  light. 

FIG.   529. —  Ditto  in  diagonal  position. 

FIG.  530. — Calcite  plate.  Section  cut  at  an  angle  of  67  1/2°  with  the  optic  axis.  Nicols  crossed, 
optic  axis  lying  in  diagonal  position;  sodium  light. 

FIG.  531. — Calcite  plate,  section  cut  parallel  to  the  optic  axis.  In  sodium  light,  between  crossed 
nicols.  Optic  axis  in  diagonal  position.  (Facing  Page  418.) 


ART.  359]  OBSERVATIONS  BY  CONVERGENT  LIGHT  419 

or  the  stage  rotated,  neither  bars  nor  circles  of  the  interference  figure  show 
change  in  position.  If,  however,  the  axis  of  the  crystal  is  inclined  to  the 
plane  of  the  section,  the  case  becomes  somewhat  different.  The  line  along 
which  there  is  no  double  refraction  is  now  no  longer  in  the  center,  but  is 
displaced  to  one  side.  On  either  side,  at  unequal  distances,  are  the  positions- 


Pic.  532.  FIG.  533.         FIG.  534.  FIG.  535-  FIG.  536.         FIG.  537. 

FlGS.   532  to  537. — Uniaxial  interference  figure.     Section  inclined  to  the  optic  axis,  which   emerges 
between  the  center  and  the  edge  of  the  field  of  view. 

of  one  wave  length  retardation.  The  figure  seen  under  the  microscope 
(Fig.  528)  is  now  not  symmetrically  placed  in  the  center  of  the  field  but  lies 
to  one  side,  and  the  rings  aa',  bbf,  etc.,  form  curves,  which  are  not  perfectly 
true  circles,  around  it.  As  the  stage  of  the  microscope  is  rotated,  the  optic 
axis  describes  a  cone.  The  interference  figure  has  moved,  by  a  rotation  to 


FIG.  538.  FIG.  539.  FIG.  540.  FIG.  541.  FIG.  542.  FIG.  543- 

FIGS.  538  to  543. — Uniaxial  interference  figure.     Section  inclined  to  the  optic  axis,  which  emerges 

beyond  the  field  of  view. 

the  right,  successively  through  the  positions  shown  in  Figs.  532  to  537 
(Cf.  also  Figs.  528-529).  If  the  inclination  of  the  axis  is  still  greater,  so  that 
it  emerges  beyond  the  field  of  view,  the  successive  positions  are  as  shown  in 
Figs.  538  to  543  (Cf.  also  Fig.  530). 

When  the  inclination  is  slight,  the  bars  remain  practically  parallel  to  the 
vibration  planes  of  the  nicols,  and  the  center  of  the  cross  moves  in  the  same 
direction  as  that  in  which  the  stage  was  rotated.  If  the  inclination  of  the 
optic  axis  is  great,  the  bar  may  appear  somewhat  curved  (Fig.  530). 1 

359.  Sections  Parallel  to  the  Optic  Axis.2 — Sections  parallel  to  the 
optic  axis  show  hyperbolae  which  are  very  similar  to  those  seen  in  biaxial 
figures  except  that  they  do  not  appear  in  the  field  until  the  stage  is  nearly 
in  the  90°  position;  then  they  move  in,  from  a  cross,  which  is  seldom  sharp, 
and  very  rapidly  move  out  again  in  the  direction  of  the  principal  axis.  Dur- 
ing the  greater  part  of  the  rotation,  therefore,  they  do  not  appear  in  the  field 

1  Cf.  Art.  372. 

2  Cf.  Art.  3/3 


420 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  360 


of  the  microscope.  In  Fig.  531,  which  is  that  of  an  interference  figure  in  a 
section  parallel  to  the  optic  axis,  the  bars  are  quite  sharp,  the  section  being 
very  thick. 

The  optic  axis  lies  in  the  direction  toward  which  the  hyperbolae  leave  the 
field.  Its  position  may  also  be  determined  by  the  appearance  of  the  inter- 
ference color.  If  the  stage  is  turned  45°  from  the  position  in  which  the  bars 

form  a  cross,  there  will  appear  a  certain 
interference  tint  in  the  center  of  the 
field.  Outward  from  this  center  there 
will  be  a  fall  in  the  color  scale  toward 
the  quadrants  in  which  the  optic  axis 
lies,  and  a  rise  in  the  other  two  quad- 
rants. This  holds  good  for  both  posi- 
tive and  negative  crystals  (Figs.  544- 
545).  In  thick  plates,  or  in  minerals 
having  high  double  refraction,  this  rise 
or  fall  in  the  interference  colors  is  observable  only  at  the  center;  beyond  that, 
there  is  a  uniform  rise. 


FIG.  544.  FIG.  545. 

FIGS.  544  and  545. — Interference  figures  of 
uniaxial  minerals  cut  parallel  to  the  optic  axes. 
Fig.  544-  Quartz  (  +  ).  Fig.  545-  Apatite  (-). 


FIG.  546. — Cones  of  equal  retardation  in 
biaxial  interference  figures. 


BIAXIAL  CRYSTALS 

360.  Sections  Cut  at  Right  Angles  to  the  Acute  Bisectrix.1 — Vibrations 
take  place  with  equal  ease  in  every  direction  about  the  optic  axes  of  a  biaxial 
crystal ;  in  consequence  there  will  be  no 
double  refraction  at  their  points  of 
emergence,  since  there  is  no  interference. 
At  these  points  (OO,  Fig.  546)  the  sec- 
tion will  appear  dark.  At  some  dis- 
tance in  every  direction  from  the  optic 
axes,  there  will  be  points  at  which  the 
retardation  is  exactly  one  wave  length. 
These  points  lie  in  an  imaginary  cone  having  an  oval  base  (Ma,  Fig. 
546).  Another  cone  will  be  the  surface  of  2\  retardation  (Mb),  another  of 
3\,  and  so  on.  Since  there  are  two  optic  axes,  there  will  be  two  such  sets  of 
cones,  and  around  them  the  ovals  unite  to  form  lemniscate  curves,  whereby, 
in  monochromatic  light,  there  will  be  produced  an  interference  figure  (Fig. 
548)  showing  two  dark  spots  surrounded  by  dark  oval  or  lemniscate  curves. 
These  dark  spots,  which  show  the  points  of  emergence  of  the  optic  axes,  are 
often  called  "eyes."  The  term  melatope  (/xeAa,  dark,  black;  and  TOTTOS, 
place)  is  here  suggested  instead.  As  in  uniaxial  figures,  and  for  the  same 
reason,  so  in  these,  also,  the  lemniscate  curves  become  curves  of  color  in 
white  light. 

1  Cf.  Arts.  368  and  374. 


FIG.  Soi.  Fro.  552. 

FIG.  54". — Interference  figure  of  titanite.  Plate  cut  at  right  angles  to  the  acute  bisectrix.  Between 
crossed  nicols  by  sodium  light,  22  1/2°  rotation  from  the  position  parallel  to  the  principal  sections  of  the 
nicols. 

FIG.  548.— Ditto.     Rotated  45°. 

FIG.  S49-— Ditto.     Rotated  90°. 

FIG.  550. — Cane  sugar.  Plate  parallel  to  100.  Interference  figure  by  sodium  light,  in  diagonal 
position. 

FIG.  551. — Diopside.     Plate  parallel  to  100  in  sodium  light.     Diagonal  position. 

FIG.  552. — Euclase.     Cleavage  plate  parallel  to  the  axial  plane  (oio).     Interference  figure  by 
sodium  light,  in  diagonal  position.  (Facing  Page  420.) 


ART.  360]  OBSERVATIONS  BY  CONVERGENT  LIGHT  421 

Besides  these  curves  of  equal  retardation  (isochromatic  curves),  there 
appear,  also,  certain  dark  bars  or  brushes  called  isogyres  («n>s,  equal  ;yvpos, 
circle).  On  rotating  the  stage,  the  cross  (Fig.  549),  which  appears  when  the 
vibration  directions  in  the  mineral  are  parallel  to  those  of  the  nicols,  dissolves 
into  two  hyperbolae  whose  poles  are  the  loci  of  the  optic  axes.  These  bars- 
revolve  in  the  opposite  direction  from  the  stage  (Figs.  547-549).  The 
smaller  the  axial  angle,  the  nearer  together  will  be  the  loci  of  the  optic  axes,1 
until,  as  a  limiting  case,  the  form  is  that  of  the  uniaxial  interference  cross. 


FIG.  553. — Vibration  directions  of  light  producing  a  biaxial  interference  figure. 

The  explanation  is  analogous  to  that  given  for  the  dark  cross  in  basal 
sections  of  uniaxial  crystals.  In  the  latter  the  two  directions  of  vibration, 
into  which  the  ray  entering  from  below  was  broken  up,  were  those  of  the  radii 
and  the  tangents.  In  a  biaxial  crystal  the  vibration  directions  are  likewise 
normals  and  tangents  to  the  advancing  wave  fronts.  The  loci  of  the  optic 
axes  are  the  foci  of  ellipses  formed  by  the  advancing  wave  front,  and  the 
normal  to  this  wave,  at  any  point,  is  the  bisectrix  of  the  angle  between  the 
two  lines  connecting  this  point  and  the  foci  (Fig.  553).  Having  determined 
the  vibration  directions  for  every  point,  the  dark  brush  can  be  readily  deter- 
mined. Applying  the  construction  of  Fig.  525  to  Fig.  553,  we  see  that  there 
will  be  darkness  wherever  the  vibration  directions  of  the  crystal  are  parallel 
to  the  principal  planes  of  the  nicols.  The  variation  of  the  positions  of  these 
dark  brushes  upon  rotating  the  stage  is  well  brought  out  in  a  diagram  given 

1  Cf.  Figs.  556  and  560.  In  the  former  the  apparent  axial  angle  is  30°,  in  the  latter 
80°. 


422 


MANUAL  OF  PETROGRAPIIIC  METHODS 


[ART.  360 


-,. 

x  x  x 
xx  xxx; 
*  x  x  x  xx 
*xxxxxx 


X.XXXXXXX-Y-V  •"/•»<  XX  XXX** 
£•/.  /X  X  X  XX  X  Ac  4--f  X  XX  XX 


by  ten  Siethoff1  (Fig.  554).  In  this  figure  the  vibration  directions  for  many 
rays  are  shown  by  small  crosses.  If  the  diagram  is  placed  upon  a  rectangular 
table,  whose  sides  may  be  taken  to  represent  the  principal  planes  of  the 
nicol  prisms  (and  consequently  the  cross  hairs  of  the  microscope),  and  it  is 
rotated  in  azimuth,  the  form  of  the  interference  figure  at  any  instant  may  be 
seen  by  observing  the  small  crosses  whose  arms  are  parallel  to  the  sides  of  the 
table.  A  rotation  of  the  diagram  through  67  1/2°  will  bring  about  the  con- 
secutive changes  of  the  figure  shown  in  Figs.  547-549.  Ten  Siethoff's  dia- 
gram also  brings  out  clearly  the  fact 
that  when  the  plane  of  the  optic  axes  is 
parallel  to  one  of  the  nicols,  the  dark 
cross  has  one  broad  and  one  narrow  bar 
(Gf.  Figs.  549,  555  and  557),  the  width 
of  the  former  depending  upon  the  optic 
angle.  In  using  this  diagram,  the  parallel 
crosses  may  be  seen  best  by  placing  the 
eye  at  one  side  and  but  a  few  centimeters 
above  the  plane  of  the  paper,  or  by  laying 
over  it  a  transparent  piece  of  celluloid 
ruled  into  rectangles. 

The  number  of  isochromatic  rings 
seen  in  a  biaxial  interference  figure, 
around  each  axis,  depends,  as  it  does  in 
uniaxial  figures,  upon  the  strength  of 
the  double  refraction  of  the  mineral,  and 
upon  the  thickness  of  the  section  (Figs. 
555  and  557).  The  number  of  complete  rings  corresponds  to  the  num- 
b^  of  wave  lengths  retardation.  This  may  be  seen  clearly  by  examining 
t*'^'  te  *"iwnce  figures  produced  by  sheets  of  mica  of  different  thicknesses, 
especially  well  by  the  different  steps  of  a  von  Fedorow  wedge.  The  first 
step,  which  has  a  retardation  of  1/4  X,  gives  a  figure  (Fig.  561)  in  which  one 
1'  mniscate  curve  completely  surrounds  the  melatopes  and  a  second  partial 
ellipse  shows  beyond  it.  The  second  step,  having  a  retardation  of  1/2  X, 
shows  the  lemniscate  curves  closing  up  toward  the  acute  bisectrix  (Fig.  562). 
The  curves  approach  each  other  still  more  in  the  third  step  with  3/4  X  retarda- 
tion, and,  when  the  retardation  is  a  single  wa^re  length  (Fig.  563),  the  first 
curve  just  unites  at  the  center  and  forms  a  figure  eight,  one  loop  around 
each  optic  axis.  The  sixth  step  (Fig.  564)  gives  a  retardation  of  i  1/2  X. 
Here  the  first  curve  is  divided  into  two,  one  forming  a  closed  curve  around 
each  axis,  and  the  second  forming  a  lemniscate  about  the  two.  The  eighth 
step  shows  a  retardation  of  two  wave  lengths  and  presents  two  complete 

1  E.  G.  A.  ten  Siethoff:    Eine  einfache  Construction  des  sogen.  Interferenzkreuzes  der 
zweiaxigen  Krystalle.     Centralbl.  f.  Min.,  etc.,  1900,  267-269. 


xxxxxx 

XXX  XX 

*X  X  x  X  Xy-V--/-  -f  4-  \-4-ArAr  AC*  X  X  X  X 


PIG.  554. — Ten  Siethoff's  diagram  show- 
ing vibration  directions  in  the  interference 
figures  of  biaxial  crystals. 


FIG.  550.  Pin.  560. 

FiC.  555. — Aragonite.  Plate  cut  at  right  angles  to  the  acute  bisectrix  in  sodium  light  between 
rossed  nicols.  Plate  1/2  mm.  thick.  Parallel  position. 

FIG.  556. — Ditto.     In  diagonal  position. 

FIG.  557- — Ditto.     Plate  2  mm.  thick.     Parallel  position. 

PIG.  558. — Ditto.     Plate  2  mm.  thick.      Diagonal  position. 

PIG.  559- — Muscovite.  Plate  at  right  angles  to  the  acute  bisectrix.  Sodium  light,  nicols  crossed. 
^^•d  positiori.  Retardation  two  wave  lengths. 

FIG.  s6o.-*-Ditto.     Diagonal  position.  (Facing  Page  422.) 


ART.  362]  OBSERVATIONS  BY  COXVERGEXT  LIGHT  423 

rings,  one  within  the  other,  about  each  axis.  (Fig.  565.  See  also  Figs. 
559-560.)  The  inner  ring  is  approximately  a  circle,  while  the  outer  is  like 
that  obtained  with  a  retardation  of  one  wave  length.  The  tenth  step  gives 
2  1/2  X  retardation,  and  the  interference  figure  is  made  up  of  two  closed  rings 
about  each  axis;  the  two  pairs  enclosed  by  lemniscate  curves  (Fig.  566).  It 
is  to  be  noted  that  there  is  no  change  in  the  distance  between  the  melatopes,1 
the  axial  angle,  of  course,  remaining  the  same. 


FIG.  561.  FIG.  562.  FIG.  563-          PIG.  564-  FIG.  565.  FIG.  566. 

FIGS.  561  to  566. — Interference  figures  in  mica  wedge,  showing  retardations  of  1/4,  1/2,  i,  i  1/2,  2- 

and  2  1/2  wave  lengths. 

361.  Sections  Cut  at  Right  Angles  to  the   Obtuse  Bisectrix.2 — When 
the  axial  angle  is  nearly  90°,  the  interference  figure  produced  in  a  section 
cut  at  right  angles  to  the  obtuse  bisectrix  resembles  that  in  a  section  cut  at 
right  angles  to  the  acute  bisectrix.     The  melatopes,  however,  will  not  appear 
in  the  field  of  view,  since  the  angular  aperture  of  the  condensers  of  most 
microscopes  will  permit  the  full  figure  to  appear  only  when  27  is  less  than 
about  60°. 

The  fact  that  the  isogyres  always  have  their  convex  sides  toward  the 
acute  bisectrix  when  the  plane  of  the  optic  axes  forms  an  angle  of  45°  with  the 
principal  sections  of  the  nicols  cannot  be  used  to  determine  whether  the  acute 
or  obtuse  bisectrix  is  in  the  field  of  view,  since  neither  brush  appears  in  the 
field  in  this  position.  When  the  obtuse  optic  angle  is  large,  it  is  generally 
possible  to  recognize  it  by  the  fact  that  the  brushes  remain  in  the  field  of  the 
microscope  but  a  short  time  upon  rotating  the  stage,  coming  in  when  the 
rotation  of  the  stage  has  brought  the  plane  of  the  optic  axes  and  the  principal 
section  of  the  nicols  close  together,  and  disappearing  immediately  after  that 
position  has  been  passed.  The  method  of  determining  the  value  of  the  optic 
axial  angle  by  this  means  is  discussed  in  full  in  Art.  416. 

362.  Sections  Inclined  to    the    Bisectrices.3 — More  and  more  of  one 
melatope  and  less  of  the  other  is  seen  as  the  section  is  more  and  more  inclined 
(Figs.  550-551).     The  convex  side  of  the  hyperbola  is  always  turned  toward 
the  acute  bisectrix  when  the  mineral  is  turned  in  the  45°  position,  and  the 
arm  rotates  in  a  direction  opposite  to  that  in  which  the  stage  is  turned  (Figs. 
567-586).     When  the  melatope  lies  near  the  edge  of  the  field  of  view,  the 

1  Cf.  Figs.  556  and  558. 

2  Cf.  Art.  375- 

3  Cf.  Art.  378,  infra. 


424 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  363 


appearances  in  uniaxial  (Fig.  530)  and  biaxial  (Fig.  551)  crystals  are  very 
similar. 

363.  Sections  at  Right  Angles  to  an  Optic  Axis.1 — Sections  cut  at  right 
angles  to  an  optic  axis  show  nearly  circular,  concentric  curves  crossed  by 


FIG.  567.  FIG.  568.  FIG.  569.  FIG.  570.  FIG.  571. 

FIGS.  567  to  571. — Biaxial  interference  figure.     Section  somewhat  inclined  to  the  plane  of  the  optic 
axes.     One  optic  axis  emerges  within  the  field  of  view,  the  acute  bisectrix  just  beyond. 


FIG.  572.  FIG.  573.  FIG.  574.  FIG.  575.  FIG.  576. 

FIGS.  572  to  576. — Biaxial  interference  figure.  Section  inclined  at  a  greater  angle  to  the  plane  of 
the  optic  axes  than  in  preceding  case.  The  optic  axes  and  the  bisectrices  emerge  beyond  the  field  of 
view. 


FIG.  577.  FIG.  578.  FIG.  579.  FIG.  580.  FIG.  581. 

FIGS.  577  to  581. — Biaxial  interference  figure.  Section  at  right  angles  to  the  plane  of  the  optic 
axes.  One  optic  axis  emerges  within  the  field  of  view,  the  acute  bisectrix  emerges  just  beyond.  The 
isogyre  is  straight  when  it  passes  through  the  center  of  the  field. 


FIG.  582.  FIG.  583.  FIG.  584.  FIG.  585.  FIG.  586. 

FIGS.  582  to  586. — Biaxial  interference  figure.  Section  at  right  angles  to  the  plane  of  the  optic 
axes.  The  optic  axes  and  the  bisectrices  emerge  beyond  the  field  of  view.  The  isogyre  is  straight 
when  it  passes  through  the  center  of  the  field. 

a  single  dark  bar,  which  is  straight  whenever  it  is  parallel  to  the  planes  of 
vibration  of  the  nicols  (Fig.  587).     Upon  rotating  the  stage,  this  bar  generally 
changes  to  a  slightly  curved  hyperbola  (Fig.  588)  with  its  convex  side  toward 
1  Cf.  Art.  377. 


FIG.  591.  FIG.  592. 

FIG.  587. — Topaz.     Section  cut  at  right  angles  to  an  optic  axis.     Xicols  crossed.     Parallel  position. 
FIG.  588. — Ditto.     Section  thicker  than  preceding.     Diagonal  position.     3V  approximately  60°. 
FIG.  589. — Andalusite.     Plate  at  right  angles  to  an  optic  axis.     Diagonal  position.     2V  =  83°  30', 
FIG.  590. — Quartz.     Plate  at  right  angles  to  the  optic  axis,     i  mm.  thick. 
FIG.  591. — Calcite.     Plate  i  mm.  thick,  cut  at  right  angles  to  the  optic  axis. 
FIG.  592. — Calcite.     Plate  3  mm.  thick.  (Facing  Page  424.) 


A RT.  :-5t ).') |  OBSER  \ '.  1  TI( ).V.V  />  1 '  COX  \  'ERG EX T  LIGH T  425 

the  acute  bisectrix.  The  amount  of  curvature  depends  upon  the  value  of 
the  optic  axial  angle.  The  smaller  the  angle,  the  greater  the  curvature. 
Figs.  587-588  show  interference  figures  of  topaz,  with  2V  approximately 
equal  to  60°.  When  2V  equals  90°  the  bar  is  straight.  It  is  generally 
impossible  to  recognize  the  curvature  when  2V  is  greater  than  80°,  as  for 
example  in  andalusite  with  2V  equal  to  83°  30'  (Fig.  589).  Sometimes  a 
bar  will  appear  approximately  straight  on  one  side  and  concave  on  the  other 
(Fig.  588).  In  such  cases  the  straight  side  is  toward  the  acute  bisectrix. 

Since  light  is  dispersed  in  all  biaxial  crystals,  a  section  can  be  actually 
at  right  angles  to  an  optic  axis  only  for  a  given  color.  The  dispersion  is 
generally  so  slight,  however,  that  it  may  be  overlooked,  and  one  will  see, 
in  white  light,  a  series  of  colored  rings  whose  tints  will  differ  from  the  pure 
colors  of  Newton's  scale  more  and  more  with  increasing  dispersion. 

364.  Sections  Parallel  to  the  Plane  of  the  Optic  Axes.1 — Sections  cut 
parallel  to  the  plane  of  the  optic  axes  (perpendicular  to  the  optic  normal  b) 
may  be  recognized  in  parallel  polarized  light  by  the  fact  that  they  show  the 
highest  interference  colors  of  any  section  of  that  mineral.     In  convergent 
light  they  show  figures  (Fig.  552)  similar  to  those  shown  by  uniaxial  crystals 
(Fig.  531)  parallel  to  the  optic  axis.     Upon  rotating  the  stage,  the  hyperbolae 
come  in  from  the  sides  very  rapidly,  darken  the  field,  and  with  very  little 
farther  rotation  immediately  disappear  in  the  direction  of  the  acute  bisectrix. 
When  the  field  is  dark  the  axes  a  and  c  are  parallel  to  the  cross  hairs. 

Becke2  has  shown  that  the  acute  bisectrix  in  this  section  is  the  line 
uniting  the  quadrants  containing  the  lower  colors.  In  negative  minerals  it  is 
a,  and  in  positive,  c.  When  the  axial  angle  approaches  90°  the  color  varia- 
tion becomes  indistinct;  when  2V  =  go°  it  disappears.3 

PROBLEM 

Use  the  gypsum  plate  as  a  mineral  section  and  determine,  by  this  method, 
the  direction  of  c. 

LOCATING  THE  POINT  OF  EMERGENCE  OF  AN  OPTIC  Axis 

365.  Uniaxial  Crystals. — If  the  point  of  emergence  of  the  optic  axis  of  a 
uniaxial  crystal  lies  within  the  field  of  the  microscope,  its  position  is  readily 
determinable  by  the  fact  that  it  lies  at  the  intersection  of  the  dark  bars. 
The  inclination  of  the  optic  axis  to  the  axis  of  the  microscope,  consequently 
the  inclination  of  the  section,  may  be  determined  by  measuring,  from  the 

1  Cf.  Art.  376. 

2  F.  Becke:    Unterscheidung  von  optisch  +  und  —  zweiaxigen  Miner  alien  mil  dem  Mi- 
kroskop.     T.  M.  P.  M.,  XVI  (1896-97),  181. 

3  Compare   the   method   given   for   the   same   determination    in  uniaxial  crystals  in 
Art.  359. 


426 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  366 


center  of  the  field,  the  distance  to  the  point  of  emergence,  or  by  measuring 

half  the  distance  between  the  cross 
in  two  positions  180°  apart,  comput- 
ing the  angular  value  by  Mallard's 
formula,  and  reducing  to  the  true 
angle  of  inclination  by  the  formula 
sin  E  =  n  sin  V. 

366.  Biaxial  Crystals.— The 

points  of  emergence  of  the  optic 
axes  in  biaxial  crystals  may  be  de- 
termined by  locating  the  point  of 
rotation  as  in  the  method  suggested 
by  Becke.1  Another  method  is  that 
of  Viola2  which  is  based  upon  the 
fact  that  the  directions  of  vibration 
in  the  isogyres  are  parallel  to  the 
vibration  planes  of  the  nicols.  If  a 
section  of  a  biaxial  mineral,  giving 
an  interference  figure  showing  the 
point  of  emergence  of  one  of  the 
optic  axes,  is  placed  upon  the  stage 
N  of  the  microscope,  and  above  it  is 
placed  a  section  of  quartz  giving  a 
uniaxial  interference  figure,  the  only 
points  of  darkness  will  be  where  the 
isogyres  intersect,  since  only  here  will 
the  rays  reaching  the  eye  be  parallel 
to  the  vibration  planes  of  the  nicols, 
and  one  will  see  two  black  dots  sur- 
rounded, in  white  light,  by  colored 
curves. 

If  I  (Fig.  593)  is  the  isogyre  from 
the  lower  thin  section,  and  Q'Q"  that 
from  the  thin  section  of  quartz,  i 
and  i  will  be  the  two  black  spots 
which  appear  where  the  two  inter- 
sect. If  II  is  the  biaxial  isogyre,  the 

1  F.  Becke:  Bestimmung  kalkreicher  Pla- 
gioklase  durch  die  Inlerferenzbilder  von  Zwill- 
ingen.  T.  M.  P.  M.,  XIV  (1894-95),  415- 
442.     Cf.  Art.  418.  infra. 

2  C.  Viola:    Methode    zur    Bestimmung 
der  Lage  der  optischen  Axen  in  Diinnschlifen.     T.  M.  P.  M.,  XV  (1896),  481-486. 


FIG.  595. 


ART.  366]  OBSERVATIONS  BY  CONVERGENT  LIGHT  427 

intersection  at  2  forms  a  single  spot.  If  the  isogyre  is  at  III,  there  will 
again  appear  two  spots,  3,  3. 

If  the  stage  of  the  microscope  is  rotated,  the  points  of  emergence  of  the 
optic  axes  of  both  the  biaxial  mineral  and  the  quartz  likewise  rotate,  but 
retain  their  relative  positions.  Thus,  in  Fig.  594,  upon  rotating  the  stage, 
the  black  dots  i,  i  become  2,  2,  then  coincide  in  3,  separate  to  4,  4,  5,  5,  etc., 
farther  and  farther  apart  as  the  stage  is  revolved.  At  the  same  time  the 
progressive  positions  of  the  melatope  of  the  quartz  are  qi9  q^  #3,  #4,  etc.  At 
3,  where  but  a  single  dark  spot  appears,  it  is  evident  that  the  optic  axis  of 
the  quartz  coincides  with  the  biaxial  isogyre. 

The  phenomenon  appears  much  simpler  if  the  nicols  are  rotated  instead 
of  the  stage.  Let  A,  Fig.  595,  be  the  point  of  emergence  of  the  optic  axis  of 
the  biaxial  mineral.  If  only  the  biaxial  mineral  section  lies  on  the  stage,  the 
isogyre  will  appear  as  a  straight  line  when  the  plane  of  the  optic  axes  lies 
parallel  to  the  vibration  direction  of  one  of  the  nicols.  If  there  is  now 
placed  above  the  biaxial  mineral  a  quartz  plate  in  such  a  position  that 
the  point  of  emergence  of  its  optic  axis  lies  on  this  line,  the  straight  bar, 
parallel  to  one  of  the  nicols,  will  still  appear,  since  along  that  line,  in  both 
minerals,  the  light  is  extinguished.  Let  the  center  of  the  quartz  cross  appear 
at  i.  If  the  nicols  are  rotated  simultaneously,  the  isogyre  of  the  biaxial 
mineral  will  assume  successively  the  positions  shown  by  the  dotted  lines. 
The  center  of  the  uniaxial  cross  of  the  quartz  will  retain  its  position,  but 
the  bars  will  revolve  so  that  they  remain  constantly  parallel  to  the  nicols. 
As  a  result,  the  points  of  intersection  of  the  two  figures  will  appear  as  two 
black  dots  (in  Fig.  595  one  of  the  dots  lies  beyond  the  field),  which  will 
rotate  about  A  as  the  nicols  are  turned.  The  spots  will  lie  nearer  to  A  than 
2  or  farther  away  than  3  according  to  whether  the  axis  of  the  quartz  lies 
nearer  or  more  distant  from  A.  If  the  axis  of  the  quartz  corresponds  with 
the  axis  of  the  biaxial  mineral,  only  a  single  black  dot  will  appear,  and  it 
will  retain  its  position  upon  rotating  the  nicols. 

To  determine  the  positions  of  the  melatopes  of  a  biaxial  mineral  as  well 
as  their  angular  distances  from  the  axis  of  the  microscope,  Viola  had  cut  a 
series  of  ten  thin  sections  of  quartz,  each  differing  by  10°  from  the  preceding 
in  its  inclination  to  the  optic  axis.  These  quartz  sections  were  so  mounted 
on  a  glass  slip  that  the  c  axis  of  all  lay  in  the  same  plane.  For  the  deter- 
mination of  the  position  of  the  optic  axis  of  a  biaxial  mineral  they  are  inserted 
successively  above  it,  but  always  in  such  a  position  that  the  dark  bar  falls 
within  the  field.  As  each  different  slice  appears,  the  nicols  are  slightly 
rotated,  and  notice  is  taken  as  to  whether  the  dark  spot  moves  or  is  stationary. 
When  it  is  stationary  the  optic  axes  of  the  two  must  coincide,  and  the  un- 
known optic  axis  forms  an  angle  with  the  axis  of  the  microscope  equal  to 
the  known  angle  of  the  quartz. 

Usually  no  quartz  slice  over  the  biaxial  interference  figure   will  produce 


428  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  366 

a  spot  absolutely  stationary,  and  it  will  then  be  possible  to  determine  only 
the  angle  as  lying  intermediate  between  those  of  two  known  quartz  sections. 
Closer  approximation  may  be  reached  if  quartz  slices  cut  at  5°  intervals  are 
used. 

If  a  rotating  stage  is  used,  instead  of  a  microscope  with  simultaneously 
rotating  nicols,  the  black  spot  of  coincidence,  of  course,  always  rotates,  but 
the  distance  of  this  spot  from  the  center  remains  constant. 


CHAPTER  XXX 
ISOTAQUES,  SKIODROMES,  AND  ISOGYRES 

367.  Isotaques  or  Curves  of  Equal  Velocity. — The  positions  of  the  isogyres 
in  random  sections  may  be  determined  by  a  method  suggested  by  Becke,1 
who  made  use  of  the  curves  of  equal  velocity,  named  by  him  isotaques  ( uros, 
equal,  ra^os,  swift,  or  to-o-Ta^s,  equally  swift).  These  had  long  previously 
been  worked  out  by  Beer,2  who  showed  that  it  is  possible,  by  following  the 
same  law  as  that  by  which  an  ellipse  is  constructed  about  its  focii — namely 
that  the  sum  of  the  distances  of  any  point  from  the  focii  is  a  constant — to 
draw  on  the  surface  of  a  sphere  two  systems  of  curves  about  two  points. 
If  2  a  is  the  major  axis  of  the  spherical  ellipse,  and  <p  and  <p'  are  the  angles 
between  the  focii  and  the  point,  then  <p-\-<pf  =  2a=  a  constant. 

If  two  series  of  spherical  ellipses  are  constructed  upon  the  surface  of  the 
sphere  about  the  points  of  emergence  of  the  optic  axes,  one  series  will  have 
for  its  center  the  acute  bisectrix,  and  the  other,  the  obtuse.  Together  the 
two  series  will  cover  the  sphere  with  a  network  of  lines,  intersecting  at  right 
angles,  by  the  aid  of  which  the  vibration  directions  at  any  point  of  the  sur- 
face may  be  easily  determined. 

Beer  showed  that  all  lines  drawn  from  the  center  of  the  sphere  to  points 
on  the  same  spherical  ellipse  represent  the  direction  of  propagation  of  rays 
having  the  same  velocity.  In  other  words,  the  ellipses  represent  curves  of 
equal  velocities.  Also,  the  vibration  directions  of  these  rays  lie  at  right  angles 
to  their  respective  ellipses,  whereby  the  tangents  .at  the  points  of  intersection 

1  F.  Becke:  Optische  Untersuchungsmethoden.  Denkschr.  Akad.  Wiss.  Wien,  LXXV 
(1904),  41  pp.* 

Idem:  Die  Skiodromen.  Ein  Hilfsmitlel  bei  der  Ableitung  der  Interferenzbilder.  T.  M. 
P.  M.,  XXIV  (1905),  1-34. 

J.  Beckenkamp :  Review  of  preceding  two.  Zeitschr.  f.  Kryst.,  XLII  (1906-7),  644- 
648. 

J.  \V.  E[vans]:  Review  ditto.     Mineralog.  Mag.,  XIV  (1907),  276-280. 

Ernst  Sommerfeldt:  Ueber  die  Bedeutung  der  Skiodromen  fur  die  Krystalloptik.  T.  M. 
P.  M.,  XXVII  (1908),  285-292. 

John  W.  Evans:  Notes  on  skiodromes  and  isogyres.  Mineralog.  Mag.,  XIV  (1907), 
230-234.  (The  movement  of  the  isogyres  with  fixed  stage  and  movable  nicols  is  described.) 

-  August  Beer:  Ueber  eine  neue  Art  die  Gesetze  der  Fortpflanzung  und  Polarisation  des 
Lichtcs  in  optisch  zvciaxigcn  Medien  darzustellen.  Grimert's  Arch.,  Th.  XVI  (1851), 
223-229. 

Idem:  Einleitung  in  die  /where  Optik.  Braunschweig,  ist  Aufl.,  1853,  401-402;  2  Aufl., 
1882,  309,  373. 

429 


430  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  368 

of  the  two  series,  being  at  right  angles  to  each  other,  represent  the  vibration 
directions  of  both  rays  propagated  along  the  radius  of  the  sphere  to  that 
point. 

The  ellipses  of  one  series  were  called  by  ecke  equatorial  ellipses  (Ge- 
schwindigkeits-ellipsen  erster  Art,  by  Beer).  They  surround  the  acute  optic 
'  angle  (2V)  and  have  values  such  that  <?-}-<?'.  >  2F  ^  180°. 

The  ellipses  of  the  other  series  were  called  by  Becke  meridian  ellipses 
(Geschwindigkeits-ellipsen  zweiter  Art,  by  Beer).  They  surround  the  obtuse 
optic  angle  (180°—  2F)  and  have  values  such  that 

<p  +  (i8o°-V)  =  2  «'  >  180°  -  2V  <  1  80°. 

In  uniaxial  crystals,  where  2  V  =  O,  the  equatorial  ellipses  become  par- 
allels, and  the  meridian  ellipses  meridians. 

Becke  further  distinguishes 
a  ellipses,  or  those  whose  tangents  represent  the  vibration  direction  of  the 

fastest  ray, 
7  ellipses,  or  those  whose  tangents  represent  the  vibration  direction  of   the 

slowest  ray, 
whereby,  in  optically  positive  (4-)  crystals 

meridian  ellipses  are  7  ellipses, 
equatorial  ellipses  are  a  ellipses  ; 
and  in  optically  negative  (  —  )  crystals 

meridian  ellipses  are  a  ellipses, 
equatorial  ellipses  are  7  ellipses. 

In  the  following  figures,  the  a  ellipses  are  shown  by  broken-,  and  the  7 
ellipses  by  dotted  lines. 

368.  Skiodromes.  —  The  isotaques,  or  curves  of  equal  velocity,  may  be 
well  shown  in  stereographic  projection.  Becke,  however,  has  given  them  by 
preference  in  orthographic  projection;  and  they  are  thus  reproduced  here. 
To  such  projections  of  the  isotaques  Becke  has  given  the  name  skiodromes 
,  a  shadow;  Spofjws,  course). 


Analytically,  the  construction  of  the  curves  is  given  in  the  work  cited  above.1 
Here  only  the  resulting  values  are  brought  together. 

Let  2  a  =  the  sum  of  the  angles  determining  the  equatorial  ellipses, 

2a'=  the  sum  of  the  angles  determining  the  meridian  ellipses, 

a     =  the  major  axis  of  any  equatorial  ellipse, 

b     =  the  minor  axis  of  any  equatorial  ellipse, 

a'    =  the  major  axis  of  any  meridian  ellipse, 

b'    =  the  minor  axis  of  any  meridian  ellipse, 

2V  =  the  acute  optic  axial  angle. 
1  Denkschriften,  etc.     Op.  cit. 


ART.  368] 


ISOTAQUES,  SKIODROMES,  AND  ISOGYRES 


431 


i.  Sections  perpendicular  to  the  acute  bisectrix  (lying  in  the  xy  plane)  (Fig.  596). 
a.  Equatorial  skiodromes  give  ellipses  wherein 


the  major  axis  a  (parallel  to  x)  =  sin  «, 


the  minor  axis  b  (parallel  to  y)  = 


cos2  V  —  cos2  a 


cos  V 


(i) 
(2) 


FIG.  596.  FIG.  597- 

FIG.  596. — Skiodrome  of  a  negative,  optically  biaxial  crystal.     Projection  of  a  section  at  right  angles 

to  the  acute  bisectrix.     Broken  lines,  skiodromes  of  the  fast  rays  (a  skiodromes) ,  dotted  lines,  skiodromes 

of  the  slow  rays  (y  skiodromes).     2F  =  6o°. 

FIG.  597. — Skiodrome   of  a   negative,   biaxial   crystal.     Projection   of   a   section   parallel   to  the 

plane  of  the  optic  axes.     2F  =  6o°. 


b.  Meridian  skiodromes  give  hyperbolae  wherein 

the  real  axis  a'  (parallel  to  x)  =  cos  «', 


\/sin2  V  —  cos2  a 
the  imaginary  axis  b  (parallel  to  y)  = 


2.  Sections  parallel  to  the  axial  plane  (lying 
in  the  xz  plane)  (Fig.  597). 

Equatorial  skiodromes  give  partial  ellipses 
wherein 

the  major  axis  a  (parallel  to  x)  =  —  —  —  »      (5)        fi 


_ 


ftj.  I    I 

7  i    'r ---'-- 


(3) 
(4) 


..VV 


the  minor  axis  c  (parallel  to  z) 


cos  a 
cosF 


•      (6) 


The   meridional    skiodromes   give   partial 
ellipses  in  which 


rt  1    1     !     1 

1      I       :        i 
l      /       '         1 

\  \ 

1      1       >         l 
\      \     .«...]  

a  r... 

-    '       '     » 

rT    I      ^        i 

V\  x      \      \       ' 
\v   \      v      -V-.-U--. 

.' 

f"J"f 

1            f 

^ii;: 

/ 

/  ' 

--/•-s 

/  r'J 

>y;& 

the  major  axis  c'  (parallel  to  z)  = 


sin  a 


(7) 


cos  V 

FIG.  598. — Skiodrome  of  a  negative  bi- 

/  axial     crystal.     Projection     of     a    section 

the  minor  axis  a'  (parallel  tO  x)    =   -\ —  —  •    (8)      perpendicular  to  the  obtuse  bisectrix. 


432  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  368 

3.  Sections  perpendicular  to  the  obtuse  bisectrix  (lying  in  the  yz  plane)  (Fig.  598). 
Equatorial  skiodromes  give  hyperbolae  wherein 

the  true  axis  c  (parallel  to  z)  =  cos  a,  (9) 

A/cos2  V  —  cos2  a 
the  imaginary  axis  b  (parallel  to  y)  =  —    : — ~ —    —  •  (io) 

The  meridian  skiodromes  give  ellipses  in  which 

the  major  axis  c'  (parallel  to  z)  —  sin  «',  (n) 


Vsin*  v  —cos2 «' 

the  minor  axis  b   (parallel  to  y)  =  —     — -. — 17 —    -  •  (12) 

sin  F 

To  construct  the  skiodromes  it  is  necessary  to  assume  successive  values  for  the 
constants  2  a  and  2  a.  In  practice  it  was  found  more  convenient  to  use  as  variables, 
not  a  and  «',  but  the  angle  which  the  short  axis  of  the  spherical  ellipse  subtends 
at  the  center  of  the  sphere.  If  this  value  is  represented  by  ft,  we  obtain,  from  the 
relationships 

COS  a     =   COS  F  COS  ft,  (13) 

and  cos  a'  —  sin  F  cos  ft,  (14) 

the  following  values  for  our  equations. 

(1)  Equatorial  skiodromes 

a=V7=^*V^*-fi,  (ia) 

b=  sin  ft.  (2a) 
Meridian  skiodromes 

a'  =  sin  F  cos  ft,  (3a) 

b'  =  tan  F  sin  ft.  (4a) 

(2)  Equatorial  skiodromes 

-cos2  F  cos2"]? 


sin  F 

c  =  cos  ft.  (6a) 
Meridian  skiodromes 

c'  =  yl  3|_£2LJ,  (7a) 

<z'  =  cos  0.  (8a) 
(3)  Equatorial  skiodromes 

c  =  cos  F  cos  0,  (pa) 

b  —  cos  F  sin  ft.  (loa) 
Meridian  skiodromes 

c'  =  Vr-sin^F^ccs2^  ^ *  *  a' 

&'  —  sin  0.  (i2a) 

The  values  of  the  equations  in  terms  of  the  angles  F  and  ft  may  be  obtained  more 

rapidly  by  first  computing  the  value  of  «  or  a  from  equations  (13)  and  (14)  and 
substituting  the  results  obtained  in  equations  i,  2,  5,  6,  7,  8,  n,  and  12.  Equations 
3a,  4a,  ga,  and  ica  may  be  used  as  they  are. 


ART.  369] 


ISOTAQUES,  SKIODROMES,  AND  ISOGYRES 


433 


The  skiodromes  shown  in  Figs.  596,  597,  and  598  were  constructed  from 
these  equations;  2V  being  taken  as  60°,  and  /8  as  15°,  30°,  45°,  60°  and  75°. 
If  the  dotted  skiodromes  in  the  figures  represent  the  vibration  directions 
of  the  slow  rays  (7  skiodromes)  and  the  broken-line  skiodromes  the  fast 
rays  (a  skiodromes),  the  crystal  represented  is  negative.  If  the  dotted 
skiodromes  represent  the  fast  and  the  broken-line  skiodromes  the  slow, 
the  crystal  is  positive.  Figures  599  and  600  show  the  skiodromes  of  sections 
at  right  angles  to  the  acute  bisectrix  and  parallel  to  the  plane  of  the  optic 
axes  when  2F  =  9o°. 

The  skiodromes,  deduced  as  above,  cover  the  field  of  view  with  curves 
which  intersect  at  right  angles  everywhere  except  near  the  edge  of  the  pro 
jection  of  the  sphere.  Here  the  light  is  no  longer  plane,  but  elliptically 


FIG.  599. — S  kiodrome  of  a  biaxial 
crystal.  2^  =  90°.  Projection  of  a  section 
perpendicular  to  a  bisectrix. 


FIG.  600. — Skiodrome  of  a  biaxial  crystal. 
2F  =  90°.  Projection  of  a  section  parallel  to 
the  plane  of  the  optic  axes. 


polarized,  and  in  the  projection  the  lines  cross  each  other  obliquely,  the 
obliquity  increasing  with  distance  from  the  center.  The  part  of  the  field  in 
the  center,  which  is  seen  in  the  conoscope,  depends  for  its  size  upon  the  mean 
refractive  index  of  the  substance  under  examination  and  the  numerical 

aperture,  of  the  conoscope.     If  2 A  =  the  numerical  aperture  and  n  the  mean 

A 

refractive  index  of  the  crystal,  —  =  the  radius  of  the  field  of  view. 

w/ 

369.  To  Construct  the  Skiodromes  for  a  Random  Section. — After  the 
skiodromes  for  the  symmetrical  sections  have  been  constructed,  it  is  a 
simple  matter  to  construct  those  for  random  sections.  Thus  if  it  is  desired 
to  construct  a  new  projection  on  a  plane  whose  pole  in  the  orthographic 
projection  is  M  (Fig.  601),  it  is  necessary  to  rotate  M  first  to  O.  The  angle 
through  which  this  point  has  been  rotated  is  shown  in  its  true  value  in  the 
section  to  the  left,  mO'o.  All  other  points  in  the  original  projection,  such  as 
A  and  B,  must  be  rotated  through  an  equal  angle.  A  vertical  plane  through 

28 


434 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  370 


A  has  for  its  trace  FE,  and  in  rotated  position  appears  as  the  circle  fae,  on 
which  a  represents  the  present  position  of  A.  The  point  a  is  now  to  be 
rotated  through  an  angle  equal  to  mO'o  or  aO'a',  which  is  most  simply  con- 


PIG.  601. — Construction  for  rotating  the  plane  of  projection  in  orthographic  projection. 

structed  by  laying  off  from  a"  to  a'"  a  chord  equal  to  mo.     All  other  points 
on  the  original  skiodrome  are  transposed  in  a  similar  manner. 

370.  Deduction  of  the  Isogyres  from  the  Skiodromes. — Every  point  in 
the  field  of  view  where  a  tangent  or  a  normal  to  a  skiodrome  lies  parallel  to 

the  principal  section  of  one  of  the  crossed 
nicols  belongs  to  an  isogyre. 

To  determine  these  points,  Becke 
used  a  sheet  of  transparent  paper  on 
which  were  drawn  rectangular  coordi- 
nates, and  which,  since  the  lines  repre- 
sent the  vibration  directions  of  the  nicol 
prisms,  may  be  called  a  nicol  net.  When 
such  a  sheet  is  placed  over  the  skio- 
dromes  (Fig.  602),  it  is  easy  to  determine 
and  connect,  by  a  continuous  line,  the 
points  where  the  two  are  parallel.  If 
FIG.  602.— skiodrome  of  an  octant  of  an  ^g  skiodrome  net  is  rotated  below  the 

inclined    section    of  a  biaxial  mineral  placed       .  . 

beneath  a  nicol  net.    /  is  the  acute  bisectrix,  nicol  net,  m  the  same  way  as  is  the  mm- 

II  the  obtuse  bisectrix,  N  the  normal,  A  an    eral    section    between    Crossed   nicols,    the 
optic  axis.     The  circle  represents  the  aperture  .   .  . 

of  the   conoscope   (ca.  30°).    The   partial  positions  of  tangency  will  change  in  the 

isogyre   of    the  meridian  skiodrome  is  shown    same    manner    as   the   isogyres    Under    the 
by  broken  lines,  that  of  the   equatorial  skio- 
drome by  dots.  microscope. 

As  may  be  seen  from  Fig.  602,  which 

shows,  beneath  a  nicol  net,  the  skiodrome  of  an  octant  of  an  inclined  sec- 
tion cut  from  a  biaxial  mineral,  there  appear  two  isogyres,  one  derived  from 


ART.  371] 


ISOTAQUES,  SKIODROMES,  AND  ISOGYRES 


435 


the  meridian  and  one  from  the  equatorial  skiodrome.  They  coincide  in  the 
center  of  the  field  and  at  the  optic  axis,  and  there  the  isogyre  forms  a 
sharply  defined  dark  bar.  Near  the  edge  of  the  field  the  skiodromes  cut 
each  other  obliquely,  and  the  vibrations  are  not  at  right  angles,  therefore 
the  isogyres  from  the  two  series  do  not  extinguish  simultaneously,  and 
the  two  curves  widely  diverge.  As  a  result,  instead  of  a  sharp  bar,  a 
diffused  brush  appears.  Practically,  it  is  sufficient  to  draw  the  mean  line 
to  represent  the  resulting  isogyre. 

The  equations  for  the  isogyres  have  been  determined  by  Hilton.1  Let  p  be  the 
radius  of  the  sphere.  Let  the  gnomonic  projection  (not  orthographic)  of  the  axes 
.Y  and  F  be  parallel  to  the  direction  of  the  nicols  and  directly  through  the  center, 


FIG.  603.  FIG.  604. 

FIGS.  603  and  604.  —  Isogyres  constructed  by  Hilton's  equation. 

and  let  the  coordinates  of  the  projections  of  the  optic  axes  be  x\yi  and  x2yz.  The 
equations  of  the  two  partial  isogyres,  which  are  the  loci  of  all  points  where  the 
projection  of  the  tangents  to  the  spherical  ellipse  are  parallel  to  the  vibration  plane 
of  the  nicols  (X,  F),  will  be 


and 


(0) 


Figs.  603  and  604  show  the  curves  for  two  positions  of  the  plane  of  the  optic 
axes,  as  developed  by  these  equations. 

(I)  SKIODROMES  OF  UNIAXIAL  CRYSTALS 

371.  Sections  Cut  at  Right  Angles  to  the  Optic  Axis. — In  sections  cut  at 
right  angles  to  the  optic  axis  (Fig.  605;  Cf.  Fig.  526),  the  isogyres  form  a 
dark  cross  along  the  vertical  and  horizontal  isotaques,  the  only  lines  in  the 
skiodrome  which  are  parallel  to  the  vibration  directions  of  the  nicols. 

1  H.  Hilton:  Ueber  die  dunklen  Biischel  von  Diinnschli/en  im  convergenten  Lichte.  Zeit- 
schr.  f.  Kryst.,  XLII  (1906-7),  277-8. 

Review  of  above  in  Mineralog.  Mag.,  XIV  (1907),  282. 


436 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  372 


372.  Sections  Inclined  to  the  Optic  Axis. — In  sections  inclined  to  the 

optic  axis  (Fig.  606;  Cf.  Fig.  528),  the  cen- 
ter of  the  cross  is  displaced  to  one  side. 
Upon  rotating  the  stage,  the  center  moves 
in  the  same  direction  (Figs.  532-537).  If 
the  inclination  of  the  section  is  slight,  all 
meridian  skiodromes  may  be  regarded  as 
straight  lines  and  all  equatorial  skiodromes 
as  concentric  circles.  Upon  rotating  the 
stage,  the  isogyres,  consequently,  are  dis- 
placed to  positions  parallel  to  the  principal 
sections  of  the  nicols,  and  approximately 

FIG.  6o5.-skiodrome  of  a  negative  Parallel  to  their   original  positions.     They 
uniaxiai  crystal  in  a  section  cut  at  right  remain  practically  straight  bars. 

angles  to  the  optic  axis.  jf  ^  indination  Qf  the  section  is   great> 

the  meridian  skiodromes  are  perceptibly 
curved,  and  the  equatorial  skiodromes  are 
no  longer  concentric.  As  a  consequence,  on 
rotating  the  stage,  the  black  bar  is  less 
rapidly  displaced  at  the  end  where  it  moves 
in  the  same  direction  as  that  in  which  the 
stage  is  rotated  (the  homodrome  end,  6/xo?? 
the  same,  Spo/^os,  course,  path)  than  in  the 
other  (the  antidrome  end,  fort,  against),  and 
it  appears  to  swing  back  and  forth.1  It 
does  not  remain  parallel  to  itself,  therefore, 
FIG.  606.— skiodrome  of  a  negative  during  the  rotation. 

In  a  uniaxiai  crystal  cut  at  an  angle 
with  the  optic  axis,  at  some  position  during 
the  rotation,  the  isogyre  forms  a  straight  bar, 
symmetrically  dividing  the  field  into  halves 
and  lying  parallel  to  the  principal  section  oj 
one  oj  the  nicols  (Figs.  532  and  538). 

373.  Sections  Parallel  to  the  Optic  Axis. 
— In  sections  parallel  to  the  optic  axis  (Fig. 
607,  Cf.  Fig.  531),  the  meridian  skiodromes 
appear  as  flattened  curves,  concave  toward 
the  center,  and  extending  from  pole  to  pole; 
the  equatorial  skiodromes,  as  parallel  lines. 
If  the  principal  sections  lie  parallel  to  the 

FIG.  607. — Skiodrome    of  a  negative       -t        ,  •  ,  ,.  <•  ,  •,  •,       ,1 

uniaxiai   crystal.     Section  parallel  to  the    Vibration  planes  of  One  of  the  niCOls,  the  in- 
optic  axis.  terference  figure  appears  as  a  broad,  black 

1  Cf.  Art.  378  for  biaxial  crystals  with  a  single  bar. 


uniaxiai  crystal, 
optic  axis. 


Section  inclined  to  the 


ISOTAQL'ES,  SKIODROMES,  AND  ISOGYRES  437 

cross,  the  outer  edges  of  the  four  quadrants  showing  a  small  amount  of 
light.  A  very  slight  rotation  of  the  nicols  will  immediately  disturb 
the  parallel  position  of  the  equatorial  skiodromes,  consequently  the  entire 
field  will  be  weakly  illuminated.  At  the  same  time  the  meridian  skiodromes 
will  cause  a  pair  of  shadowy  hyperbolae  to  appear,  which,  however,  disappear 
on  very  little  more  rotation. 

PROBLEMS 

Construct  a  nicol  net  on  transparent  paper,  and  draw  the  isogyres  for  o°,  30°, 
60°,  and  90°  rotation  of  the  mineral  section  shown  in  Figs.  605,  606,  and  607. 

Construct  the  isogyres  formed  by  rotating  the  two  nicols  simultaneously  through 
the  same  angles  as  before,  leaving  the  mineral  stationary.  Compare  the  results. 

Examine  (a)  basal  section,  (b)  inclined  section,  (c)  section  parallel  to  crystallo- 
graphic  c  of  quartz  and  of  calcite. 

II.  SKIODROMES  OF  BIAXIAL  CRYSTALS 
A.   SECTIONS  PERPENDICULAR  TO  THE  PRINCIPAL  VIBRATION  AXES 

374.  Sections  Perpendicular  to  the  Acute  Bisectrix. — In  sections  per- 
pendicular to  the  acute  bisectrix  (Fig.  596;  Cf.  Figs.  547-549), l  the  isogyres 
form  a  dark  cross  when  the  vibration  axes  are  parallel  to  the  principal  sec- 
tions of  the  nicols  (Fig.  549).    Of  the  two  dark  bars,  the  one  passing  through 
the  points  of  emergence  of  the  optic  axes  is  called  the  axial-bar  or  axial- 
isogyre,  and  is  much  more  sharply  defined  than  the  bar  at  right  angles  to 
it.     The  latter  is  called  the  central-bar  or  central-isogyre,  and  is  more  or 
less  diffused,  the  width  increasing  with  increasing  axial  angle.     When  the 
stage  is  rotated,  the  dark  cross  separates  into  two  hyperbolae  (Fig.  548), 
half  the  central  bar  uniting  with  half  the  axial  bar  to  form  each.    The  end 
which  belongs  to  the  axial  bar,  however,  is  distinguished  from  that  which 
belongs  to  the  central  bar  by  the  fact  that  it  is  homodrome  while  the  latter 
is  antidrome.     The  velocity  of  the  homodrome  end  depends  upon  the  location 
of  the  point  of  emergence  of  the  optic  axis.     If  this  lies  outside  the  field  of 
view  of  the  microscope,  the  homodrome  end  moves  more  rapidly  than  the 
rotation  of  the  stage;  if  it  lies  exactly  on  the  periphery,  the  velocities  are 
the  same;  and  if  it  lies  within  the  field,  it  moves  more  slowly.     In  every  case, 
however,  the  movement  is  in  the  same  direction  as  the  stage. 

375.  Sections  Perpendicular  to  the   Obtuse  Bisectrix. — The   angular 
aperture  of  an  ordinary  petrographic  microscope  will  permit  the  points  of 
emergence  of  both  optic  axes  to  be  seen  when  the  apparent  axial  angle  (2E)  is 
less  than  90°,  consequently  neither  melatope  can  be  seen  in  sections  cut  at 
right  angles  to  the  obtuse  bisectrix  (Fig.  598). 2    When  the  true  axial  angle 

1  Cf.  Art.  360. 
2Cf.  Art.  361. 


438  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  376 

(2V)  is  large  and  approaches  90°,  the  interference  figure  seen  in  sections  cut 
at  right  angles  to  the  obtuse  bisectrix  differs  very  little  from  that  seen  in 
sections  perpendicular  to  the  acute  bisectrix.  Which  bisectrix  is  present 
may  best  be  determined  by  the  fact  that  the  hyperbolae  of  the  interference 
figure  around  the  obtuse  bisectrix  disappear  from  the  field  with  less  rotation 
of  the  stage  than  do  those  around  the  acute  bisectrix. 1 

376.  Sections  Perpendicular  to  the  Optic  Normal. — The  isogyres  formed 
in  sections  perpendicular,  or  nearly  perpendicular,  to  the  optic  normal  b 
(Fig.  597;  Cf.  Fig.  552)  are  very  indistinct,2  the  part  of  the  skiodrome  seen 
in  the  field  of  the  microscope  presenting  a  network  of  lines  with  practically 
rectangular  intersections.     All  such  sections  are  characterized  by  the  rapid 
lighting  up  of  the  field  upon  a  very  slight  rotation  of  the  stage,  and  the  for- 
mation of  two  indistinct,  shadowy  hyperbolae,  which  move  off  with  but  little 
more  rotation. 

PROBLEMS 

With  the  nicol  net,  construct  the  isogyres  for  o°,  30°,  60°,  and  90°  rotation  of  the 
sections  indicated  in  Figs.  596,  597,  and  598. 

Compare  the  isogyres  formed  by  rotating  the  nicols  through  the  same  angles, 
leaving  the  mineral  section  stationary. 

Examine  the  interference  figures  in  (a)  the  1/4  X  mica  plate,  (b)  the  100  face  of 
topaz  or  enstatite,  (c)  the  gypsum  unit  retardation  plate. 

B.  SECTIONS  PERPENDICULAR  TO  AN  OPTICAL  PLANE  OF  SYMMETRY 

377.  Sections  Perpendicular  to  the  Plane  of  the  Optic  Axes. — Of  all 

planes  perpendicular  to  the  plane  of  the  optic  axes  (Figs.  608-609;  Cf.  Figs. 
587-589),  the  most  important  are  those  nearly  or  quite  at  right  angles  to 
an  optic  axis.3  Fig.  608  is  the  skiodrome  of  an  optically  negative  crystal, 
and  Fig.  609  of  a  neutral  crystal  with  an  axial  angle  of  90°.  In  each  case 
there  appears  but  a  single  bar,  which  remains  in  the  field  of  view  during  a 
complete  rotation.  Both  ends  are  antidrome. 

When  the  principal  vibration  directions  of  the  crystal  and  the  nicols  are 
parallel,  the  isogyre  is  a  straight  bar  which  is  parallel  to  the  principal  section 
of  one  of  the  nicols.  If  the  section  is  cut  exactly  at  right  angles  to  the  plane 
of  the  optic  axes,  the  bar,  when  straight,  divides  the  field  symmetrically; 
if  the  section  is  somewhat  inclined,  the  bar,  when  it  straightens  out,  does 
not  cross  the  center  (Figs.  567,  571,  572,  576). 

1  Cf.  the  Michel-Levy  (Art.  417)  and  the  Becke  (Art.  421)  methods  for  measuring  2E. 
~  Cf.  Art.  364. 
3  Cf.  Art.  363. 


ART.  378] 


ISOTAQUES,  SKIODROMES,  AND  ISOGYRES 


439 


The  most  advantageous  position  for  the  study  of  these  sections  is  when 
the  crystal  is  placed  in  the  diagonal  position.  When  the  true  optic  axial 
angle  is  less  than  90°,  the  isogyre  takes  the  form  of  a  hyperbola  with  the  apex 
of  the  convex  side  pointing  toward  the  acute  bisectrix. 

The  smaller  the  axial  angle,  the  sharper  will  be  the  curve,  being  a  right 
angle  when  2V  =  o°, — that  is,  in  a  uniaxial  crystal.  If  2  V  becomes  greater, 
the  curvature  of  the  isogyre  becomes  less  and 
it  flattens  out  more  and  more  until,  when 
2F  =90°  (Figs.  589  and  609),  it  is  a  straight 
bar  which  lies  at  45°  to  the  plane  of  the  optic 
axes  when  the  crystal  is  in  the  diagonal  posi- 
tion. When  2  V  is  greater  than  90°,  the  curve 
bends  in  the  opposite  direction,  that  is,  the 
acute  axial  angle  becomes  the  obtuse,  and  vice 
-versa. 

Sections  perpendicular  to  the  plane  of 
the  optic  axes,  but  intermediate  in  position 
between  those  perpendicular  to  a  bisectrix 
and  perpendicular  to  an  optic  axis,  show  a 
single  isogyre  which  is  straight  when  it  is 
parallel  to  the  principal  section  of  one  of 
the  nicols.  The  homodrome  end  shows  the 
direction  of  the  optic  axis,  and  the  antidrome 
end  that  of  the  bisectrix.  In  no  case,  except 
that  of  sections  exactly  at  right  angles  to  the 
plane  of  the  optic  axes  (Figs.  577,  581,  582, 
586),  does  the  straight  isogyre  symmetrically 
divide  the  field,  but  lies  to  one  side  of  the 
middle.  In  other  positions  the  bar  is  curved, 
and  the  homodrome  end  moves  less  rapidly 
than  the  antidrome. 


PIG.  608. — Skiodrome  of  a  nega- 
tive biaxial  crystal,  section  at  right 
angles  to  one  of  the  optic  axes.  Only 
that  portion  of  the  construction  sphere 
which  represents  the  field  of  view  of 
the  conoscope  is  shown.  Note  that 
the  convex  side  of  the  isogyre  lies  on 
the  side  toward  the  acute  bisectrix. 


C.  INCLINED  SECTIONS 


FIG.  609. — Skiodrome  of  a  neutral 
biaxial  crystal  (2^  =  90°)  in  a  section 
cut  at  right  angles  to  an  optic  axis. 
The  isogyre  is  a  straight  bar. 


378.  Random  Sections. — Among  random 
sections1  of  biaxial  minerals  in  thin  rock-slices, 
the  most  common,  of  course,  are  those  that 

are  inclined  to  the  optical  symmetry  planes,  so  that,  in  parallel  position,  the 
center  of  the  dark  cross,  in  general,  will  not  be  in  the  center  of  the  field  of  view, 

1  Fouque  et  Levy:  Mineralogie  micrographique,  Paris,  1879,  IO2-  Levy  et  Lacroix: 
Les  mineranx  des  roches,  Paris,  1888.  F.  Becke:  Zur  Unterscheidung  ein-  und  zweiachsiger 
Krystalle  im  Konoskop.  T.  M.  P.  M.,  XXVII  (1908),  177-178. 

Cf.  Art.  362,  supra. 


440 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  379 


but  will  lie  to  one  side,  and  the  isogyre  will  not  divide  the  field  symmetrically. 
The  bar  is  straight  and  lies  in  the  center  of  the  field  and  parallel  to  the 
principal  sections  of  the  nicols  only  when  the  section  is  perpendicular  to  a 
symmetry  plane  (Figs.  577,  581,  582,  586).  More  often  the  section  lies  to 
one  side,  consequently  inclined  to  all  three  symmetry  planes  of  the  optical 
ellipsoid.  In  such  a  case,  the  isogyre  will  cross  the  center  of  the  field  of  mew 
at  an  angle  to  the  vibration  planes  of  the  nicols  (Figs.  569,  575,  610.)  In  its 
straight  position  it  will  lie  to  one  side  of  the  center  (Figs.  567,  571,  572,  576). 
When  2V  =  90°,  the  bar  will  be  straight  in  every  position,  and  when  it 
crosses  the  center  it  will  make  an  angle  of  45°  (Fig.  611).  The  separation 
from  uniaxial  crystals  may  be  made  by  noting  that  upon  rotating  the  stage, 
the  homodrome  end  will  move  more  rapidly  than  the  antidrome.1 


FIG.  6 10. — Skiodrome  of  a  negative  biaxia! 
crystal  (2F  =  6o°).  Inclined  section,  between 
the  axis  and  the  normal.  Isogyre  curved. 


/     >.. 

/•/.>. 

/  "••/       / 


FIG.  6 1 1. — Skiodrome  of  a  neutral  biaxial 
crystal  (2F  =  QO°).  Inclined  section,  between 
axis  and  normal.  The  isogyre  forms  a  straight 
bar. 


PROBLEM 

Examine  inclined  sections  of  augite  and  of  olivine  showing  a  single  bar,  and  note 
the  difference  between  the  isogyres  seen  here  and  those  seen  in  an  inclined  section  of 
quartz. 

379.  Equations  for  the  Isogyres  or  Neutral  Curves. — Analytically  the 
isogyres  in  biaxial  crystals  may  be  explained  as  follows: 

(a)  Sections  at  Right  Angles  to  the  Acute  Bisectrix.  The  Line  Connecting  the 
Points  of  Emergence  of  the  Optic  Axes  Forms  an  Angle  (ft)  with  the  Principal  Sec- 
tion of  One  of  the  Nicols. — Let  Fig.  612  represent  the  isogyres  seen  in  the  inter- 
ference figure  of  a  biaxial  mineral  cut  at  right  angles  to  the  acute  bisectrix  and 
turned  to  a  diagonal  position,  so  that  the  line  O'O,  connecting  the  points  of  emerg- 
ence of  the  optic  axes,  makes  an  angle  of  0  with  t^e  principal  plane  of  the  polarizer 
P'P.  The  point  R,  lying  on  the  neutral  curve,  is  dark,  consequently  the  vibra- 
tion directions  of  the  ray  CR,  emerging  at  R,  are  bR  and  aR.  According  to  the 
Fresnel  construction,  the  vibration  direction  of  any  ray  CR  lies  in  the  plane  which 
bisects  the  angle  O'RO  in  space,2  that  is,  it  is  the  intersection  of  the  two  planes 


1  Cf.  Art.  372  for  uniaxial  crystals. 

2  See  Art.  349. 


ART.  379] 


ISOTAQUES,  SKIODRO.}fES,  AXD  ISOGYRES 


441 


ORC  and  O'RC,  each  of  which  contains  the  ray  CR  and  an  optic  axis  (CO  or  CO'}. 
For  small  angles  we  may  use  the  orthographic  projection  (Fig.  612)  instead  of  the 
angle  in  space,  whereby  the  trace  of  the  vibration  plane  is  the  line  Ra  which 
bisects  the  angle  O'RO.  Let  the  coordinates  of  the  point  of  emergence  of  the 
optic  axis  O  be  #'  and  yf,  and  those  of  the  point  R  be  #  and  v. 
From  the  figure  we  have 


cot  Rha 


cot  R0d=~°=-   —„ 

dR     y  —  y 


Rha=Rfa,  since  Ra  is  at  right  angles  to  PP'  and  bisects  the  angle  O'RO,  therefore 


y-y'    y+yf 
whereby 

x'y'  =  xy.  (i) 

This  is  the  equation  of  a  hyperbola  passing  through  O  and  0'  and  referred  to  its 

asymptotes.     The   locus  of  R,  therefore,    is 

a    rectangular    hyperbola  whose  asymptotes 

are  parallel  and  at  right  angles  to  the  prin- 

cipal sections  of  the  polarizer.     The  curve  rep- 

resents the  position  of  all  points  whose  vibra- 

tion  directions   are   parallel  to   that  of   the 

polarizer. 

In  a  similar  manner,  f  or  /3  =  90°,  two  other 
hyperbolic  branches  will  be  found,  also  passing  P'- 
through  0  and  0',  and  having  for  their 
asymptotes  lines  parallel  and  at  right  angles 
to  the  principal  section  of  the  analyzer.  These 
curves  represent  the  positions  of  all  points 
whose  vibration  directions  are  parallel  to  that 
of  the  analyzer. 

The  isogyres,  therefore,  will  consist  of  two 
rectangular    hyperbolae     (four    hyperbolic 

branches),  two  branches  passing  through  0  and  two  through  Of.  The  bars  -will 
appear  dark  when  the  nicol  prisms  are  crossed,  since  the  hyperbola,  representing  the 
light  passing  through  the  polarizer  and  parallel  to  its  principal  section,  is  exactly 
covered  by  the  hyperbola  representing  the  light  passing  through  the  analyzer 
and  parallel  to  its  principal  section.  The  two  being  at  right  angles,  all  light  is 
extinguished. 

When  the  nicols  are  paralle1,  the  hyperbolae  will  appear  light,  since  along  these 
lines  all  light  is  transmitted  without  change. 

(b)  Sections  at  Right  Angles  to  the  Acute  Bisectrix.  Line  Connecting  the  Melatopes 
Parallel  to  the  Principal  Section  of  Polarizer  or  Analyzer—  If  the  line  OO'  (Fig.  612) 
coincides  with  PP'  or  A  A',  0  =  o,  y'  becomes  o,  and  equation  (i)  becomes 

xy  =  o.  (2) 

The  hyperbola  is  reduced  to  its  asymptotes  and  forms  a  cross  (Fig.  549). 


CHAPTER  XXXI 
DISPERSION  OF  LIGHT  IN  CRYSTALS 

380.  Normal  and  Anomalous  Dispersion.  —  When  a  beam  of  white  light 
is  refracted  by  a  transparent  medium,  it  is  separated  into  rays  of  different  wave 
lengths,  consequently,  of  different  colors.  For  example,  in  passing  obliquely 
from  air  to  glass,  the  beam  of  white  light  (W,  Fig.  613)  is  separated  or  dis- 
persed1 into  colored  rays  following  the  order  of  the  spectrum.  Ordinarily 
the  ray  bent  least  from  the  direct  path  (having  the  greatest 
angle  of  refraction)  is  the  red,  and  the  one  bent  most,  the 

r 

violet,  but  in  certain  substances  the  order  is  different,  as 
<T~  ~T  was  first    discovered  by   Talbot2  about  1840,  and  redis- 

(__  //   \  _  \  covered  by  Leroux3  in  1860.     To  this  phenomenon  the  name 


anomalous  dispersion  is  given,  although  there  is  actually 
Hghter 


'          Per~  nothing   anomalous   in   the  phenomenon.     Schuster4  pro- 


posed the  term  selective  dispersion. 

As  for  the  cause  of  normal  dispersion,  it  follows,  in  isotropic  media,  very 
simply  from  the  formula 

V 

n~    y,> 

where  V  is  the  velocity  of  the  light  of  a  certain  wave  length  in  the  given 
medium.     Thus  red  waves  are  longer  than  blue,  consequently,  of  two  such 

1  For  the  theoretical  discussion  of  dispersion  see: 

L.  Lorenz:  Theorie  der  Dispersion.     Wiedem.  Ann.,  XX  (1883),  1-21. 

E.  Lommel:  Das  Gesetz  der  Rotationsdispersion.  Ibidem,  XX  (1883),  578-592. 

Paul  Drude:  The  theory  oj  optics.  Translated  by  Mann  and  Millikan.  New  York, 
1902,  Chapt.  V. 

Thomas  Preston:  The  theory  of  light.  London,  3d  ed.,  1901,  406-408;  429-430;  477- 
478;  485-488. 

Arthur  Schuster:  An  introduction  to  the  theory  of  optics.    London,  1904,  Chapt.  XL 

Robert  W.  Wood:  Physical  Optics.     New  York,  1905,  Chapt.  V. 

A.  Winkelmann:  Handbuch  der  Physik.  VI,  Optik.  Leipzig,  2  Aufl.,  1906,  618- 
636,  1316-1333. 

2H.  F.  Talbot:  Note  on  some  anomalous  spectra.  Proc.  Roy.  Soc.,  Edinburgh,  VII 
(1872),  408-410. 

P.  G.  Tait:  On  anomalous  spectra.     Ibidem,  410-412. 

Idem:  Light.     Edinburgh,  2d  ed.,  1889,  171-72. 

3  F.  P.  Leroux:  Dispersion  anomale  de  la  vapeur  d'iode.     Comptes  Rendus.,  LV  (1862), 
126-128. 

Idem:  Anomale  Dispersion  des  loddampfes.     Pogg.  Ann.,  CXVII  (1862),  659-660. 

4  Loc.  cit. 

442 


ART.  381] 


DISPERSION  OF  LIGHT  IN  CRYSTALS 


443 


rays  derived  from  the  same  beam  of  white  light  and  therefore  having  their 
vibration  periods  alike,  the  red  will  travel  farther  in  a  given  time.  It  will 
therefore  differ  less  than  the  blue  from  the  distance  traveled  by  the  same  ray 
in  air  and  will  be  less  deflected  (Fig.  613). 

In  crystals  other  than  those  that  are  isotropic,  the  difference  in  the  refrac- 
tive indices,  consequently  of  the  velocity  of  the  light  in  different  directions, 
has  its  effect  upon  the  dispersion.  For  example,  in  a  uniaxial  crystal  whose 
value  for  co  is  not  very  different  from  that  for  e,  it  may  be  that  e  is  the  direction 
of  vibration  of  the  slow  ray  for  light  of  one  color  while  for  another  color  it  is 
the  direction  of  the  fast  ray.  As  a  consequence  the  crystal  is  positive  for  the 
first  light  and  negative  for  the  second.  For  some  intermediate  color  the 
crystal  must  appear  isotropic. 

In  biaxial  crystals  the  effect  of  dispersion  is  not  so  simply  shown,  the 
phenomenon  depending  not  only  upon  the  different  values  of  the  refractive 
indices  in  different  directions,  but  also  upon  the  crystal  symmetry.  The 
result  is  that  there  may  be  a  dispersion  of  the  optic  axial  angle,  of  the  bisec- 
trices, or  of  the  axial  plane. 

DISPERSION  IN  ORTHORHOMBIC  CRYSTALS 

381.  Dispersion  of  the  Optic  Axes. — The  dispersion  of  the  optic  axial 
angle,  usually  called  the  dispersion  of  the  optic  axes,  depends  upon  the  fact 
that  the  three  refractive  indices   or,  |8,  and  7,  are 
different  for  different  colored  rays,  consequently 
I',  in  the  formula 


FIG.  614. — Dispersion  of  the 
optic  axes,      p  <  u. 


(Eq.  19,  Art.  71) 


has  different  values.  In  practice  it  is  customary 
to  express  this  difference  by  indicating  the  relation 
between  the  two  extreme  rays.  Thus  p>v  means 
that  the  angle  for  the  red  rays  (p)  is  greater  than 
that  for  the  violet  (u).  The  reverse  relationship 
is  expressed  by  p<v  (Figs.  614-615). 

Dispersion  of  the  optic  axes  is  found  in  ortho- 
rhombic  crystals.  The  three  principal  vibration 
axes  always  coincide  with  the  three  crystallographic 
axes  for  all  colors,  but  not  necessarily  do  the  same 
crystallographic  axes  coincide  with  the  same  vibra- 
tion axes  for  different  colors.  Any  one  of  the 
crystallographic  axes  may  be  parallel  to  the  maxi- 
mum, minimum,  or  intermediate  vibration  axis  for 
a  certain  color  of  light,  but  for  another  wave  length  it  may  be  parallel 
to  a  different  vibration  axis.  The  symmetry  of  the  dispersion  depends 


FIG.  615. — Dispersion  of  the 
optic  axes,     p  <v. 


444 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  382 


jopon_the  symmetry  of  the  crystal.  Since  the  principal  vibration  directions 
coincide  with  the  crystallographic  axes,  the  symmetry  of  the  crystal  will 
prevent  any  dispersion  of  the  bisectrices.  The  interference  figures  will 
be  symmetrical  to  two  planes  which  are  at  right  angles  to  each  other, 
pass  through  the  acute  bisectrix,  and  contain  the  axes  of  the  crystal. 
One  plane  is  the  plane  of  the  optic  axes,  the  other,  the  plane  at  right 
angles  to  it. 


FIG.  616.  FIG.  617. 

FIGS.  616  and  617. — Interference  figure  produced  by  dispersion  of  the  optic  axes,     p  <  v. 

The  effect  on  the  interference  figure  is  shown  diagrammatically  in  Figs. 
615-617.  Under  the  microscope  it  will  be  seen  that  the  color  whose  disper- 
sion is  least  always  lies  immediately  adjacent  to  the  hyperbolae  on  the  con- 
cave sides  (Fig.  617).  The  curves  around  the  melatopes  will  be  of  equal 
size,  and  the  colors  in  the  two  will  be  symmetrical  with  respect  to  the  plane 
of  the  optic  axes.  They  will  also  be  symmetrical  to  the  plane  through  the 
bisectrix  and  at  right  angles  to  the  plane  of  the  optic  axes. 

382.  Crossed  Axial  Plane  Dispersion. — In  a  few 

crystals  the  refractive  indices  for  light  of  different 
colors  vary  so  unequally  that,  for  example,  the  axis 
of  least  ease  of  vibration  changes  position  with  that 
of  intermediate  ease.  As  a  consequence  the  plane  of 
the  optic  axes  may  come  to  lie  at  right  angles  to  its 
former  position  and  we  have  crossed  axial  plane  dis- 
persion1 (Fig.  6 1 8).  At  some  intermediate  position, 
that  is  with  some  intermediate  color,  the  crystal 

must  be  uniaxial.     For  example,  in  brookite  (Figs.  619-624)  we  have  for 

monochromatic  light: 

1  A.  E.  H.  Tutton:  Attgemeine  Erkldrung  des  Phanomens  der  Dispersion  in  gekreuzten 
Axenebenen.  Zeitschr.  f.  Kryst.,  XLII  (1907),  554-557. 

Idem:  The  optical  constants  of  gypsum  at  different  temperatures,  etc.  Proc.  Roy.  Soc. 
London,  (A),  LXXXI  (1908),  40-57. 

Idem:  Crystallography  and  practical  crystal  measurement.    London,    1911,  784-798. 


FIG.     6 1 8. — Crossed     axial 
plane  dispersion. 


FIG.  623.  FIG.  624. 

FIGS.  619-624. — Crossed  axial  plane  dispersion  in  brookite.      Plate  cut  at  right  angles  to  the  acute 
bisectrix,  normal  position. 

FIG.  619. — Wave  length  of  source  of  light  645-620^-     Plane  of  the  optic  axes  ooi. 

FIG.  620. — Wave  length,  590-575^^-     Plane  of  the  optic  axes  ooi. 

FIG.  621. — Wave  length,  575-555M/*.     Uniaxial. 

FIG.  622. — Wave  length,  560-540^^.     Plane  of  the  optic  axes  oio. 

FIG.  623. — Wave  length,  540-520/1/1.     Plane  of  the  optic  axes  oio. 

FIG.  624. — Brookite  by  white  light.     Crossing  of  planes  of  the  optic  axes.  (Facing  Page  444.) 


ART.  384] 


DISPERSION  OF  LIGHT  IN  CRYSTALS 


445 


Light 

Wave  length 

MM 

Color 

2E 

Plane  of  optic  axes 

Li 

670 

Red  

58°    o' 

ooi  (Fig.  619) 

JJa 

580 

Yellow  

38°  10' 

ooi  (Fig.  620) 

Tl 

555 
si  e 

Yellowish  green  
Green 

o° 

21°  40' 

a  axis  (Fig.  621) 
oio  (Fig.  623) 

C2  C 

Green 

33°    o' 

OIO 

When  white  light  is  used,  the  interference  figure  is  complicated  (Fig.  624), 
and  is  the  resultant  of  the  superposed  figures  derived  from  the  different 
colored  rays. 

PROBLEM 

Examine  the  interference  figures  of  thomsonite  (p<  v,  strong),  olivine  (p<  v, 
distinct);  anhydrite  (p<  v,  strong);  sillimanite  (p>  v,  strong). 

Examine  the  interference  figure  of  a  (100)  section  of  brookite  (p  >  v)  by  red, 
yellowish  green,  green,  and  white  light,  and  note  the  difference  in  the  appearance 
of  the  interference  figures. 

DISPERSION  IN  MONOCLINIC  CRYSTALS 

383.  Dispersion  of  the  Bisectrices. — In  the  monoclinic  system  there  is 
but  a  single  plane  of  symmetry,  namely,  the  plane  perpendicular  to  the  b 
axis.     This  axis  is  the  direction  of  one  of  the  principal  vibration  axes  or 
axes  of  the  optical  ellipsoid.     It  has,  therefore,  the  same  position  for  all 
colors  of  light,  but  the  other  two  axes  may  be  dispersed  differently  in  the 
plane  of  symmetry.     This  dispersion  is  known  as  the  dispersion  of  the 
bisectrices.     It  is  always  accompanied  by  a  dispersion  of  the  optic  axes. 
Three  cases  may  occur: 

384.  Case  I.    Inclined  Dispersion  (of  Both  Bisectrices). — When  the  b 
axis  of  the  crystal  coincides  with  the  j3  axis  of  the  optical  ellipsoid,  the  plane 


FIG.  625. 


FIG.  626. 


FIG.  628. 


FIGS.  625  to  628. — Inclined  dispersion  in  monoclinic  crystals,      p  <  v. 

of  the  optic  axes  (plane  of  a  and 7)  coincides  with  the  symmetry  plane  (oio). 
Dispersion  in  such  crystals,  being  symmetrical  only  to  the  plane,  is  produced 
by  the  greater  or  less  displacement  of  the  axes  of  the  optical  ellipsoid  for 
different  colors,  consequently  the  bisectrices  of  the  optic  angles  for  different 


446  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  385 

colors  differ  by  some  angle  in  this  plane  (Figs.  625-628).  Such  dispersion 
was  called  dispersion  inclinee  by  Des  Cloizeaux.1  Since  it  is  always  accom- 
panied by  a  dispersion  of  the  optic  axes,  the  interference  figure  produced  is 
no  longer  symmetrical  with  respect  to  a  plane  at  right  angles  to  the  plane  of 
the  optic  axes,  although  it  must  necessarily  remain  so  with  respect  to  the  latter 
plane  (Figs.  625  and  627).  The  isochromatic  curves  around  the  melatopes, 
produced  by  the  displacement  of  axes  having  larger  or  smaller  angles  between 
them,  will  be  larger  and  more  elongated  at  one  melatope  than  at  the  other. 
The  colors,  likewise,  will  be  more  intense,  and  their  sequence  different.  If 
the  dispersion  of  the  bisectrices,  as  well  as  that  of  the  axes,  is  great,  the 
relations  of  the  red  and  the  violet  to  the  hyperbolae  will  be  reversed;  in  one 
the  red  will  lie  on  the  concave  side,  in  the  other  the  violet.  Usually,  how- 
ever, the  dispersion  is  too  small  to  reverse  the  order  of  the  colors,  although 
the  intensity  may  be  different,  for  example,  in  gypsum  (Fig.  628). 

PROBLEM 

Examine  the  interference  figure  of  gypsum  (P<V)  and  of  diopside  (p>u), 
first  by  white  light,  then  with  color  screens. 

385.  Case  II.  Horizontal  Dispersion  (of  the  Acute  Bisectrix). — When  /? 
and  the  acute  bisectrix  lie  in  the  plane  of  symmetry,  and  the  third  axis  of  the 
optic  ellipsoid  coincides  with  crystallographic  b,  the  plane  of  the  optic  axes 


FIG.  629.  FIG.  630.  FIG.  631. 

FIGS.  629  to  631. — Horizontal  dispersion  of  the  acute  bisectrix.     p<u. 

will  lie  at  right  angles  to  the  symmetry  plane  for  all  colors,  but  the  inclination 
of  this  plane  to  the  vertical  axis  may  differ  for  different  colors  (fig.  629^). 
The  figures,  produced  by  the  illumination  of  the  mineral  by  the  spectral 
colors  in  order,  will  successively  occupy  parallel  positions  farther  and  farther 
removed  from  the  starting  point  (Figs.  632-637).  This  dispersion,  conse- 
quently, is  called  horizontal  dispersion  (dispersion  horizontal,  Des  Cloi- 

1  A.  Des  Cloizeaux:  Sur  I'emploi  des  proprietes  optiques  birefringentes  pour  la  determina- 
tion des  especes  cristallisees.  Ann  d.  Mines,  XIV  (1858),  341-342. 

Idem:  Memoire  sur  I'emploi  du  microscope  polarisant  et  sur  I' etude  des  propriety  optiques 
birefringentes  propres  a  determiner  le  systeme  cristallin  dans  les  cristaux  naturel  ou  artificiels. 
Ibidem,  VI  (1864),  565-569. 


FIG.  636.  pIG.  037. 

FIGS.  632-637. — Effect   of   horizontal   dispersion    in    monoclinic   crystals.      Interference   figure   of 
rubidium  platino  cyanide.      Diagonal  position. 
_  FIG.  632. — Wave  length  of  source  of  light  450-460^/1- 
FIG.  633. — Wave  length  480-490^- 
FIG.  634. — Wave  length  520-530^- 
FIG.  635.— Wave  length  546.  IMM  (Green,  Hg  light). 
FIG.  636. — Wave  length  560-5  yo/iM- 
FIG.  637.— Wave  length  600-620^.  (Facing  Page  446.) 


ART.  336]  DISPERSION  OF  LIGHT  IN  CRYSTALS  447 

zeaux).1  The  interference  figure  in  white  light,  the  resultant  of  those  for  all 
the  colors,  will  still  be  symmetrical  with  respect  to  the  plane  of  symmetry, 
but  not  in  a  direction  at  right  angles  to  it  (Figs.  630-631).  This  dispersion 
is  most  clearly  seen  when  the  interference  figure  is  placed  with  its  principal 
sections  parallel  to  those  of  the  nicols  (Fig.  630).  The  dark  bar,  passing 
through  the  two  melatopes,  will  be  bordered  at  the  top  and  bottom  by 
different  colors. 

PROBLEM 

Note  the  horizontal  dispersion  shown  by  the  interference  figure  of  sanidine;  of 
muscovite. 

386.  Case  HI.  Crossed  Dispersion  (of  the  Obtuse  Bisectrix). — When 
the  acute  bisectrix  ( =  a  or  7)  coincides  with  crystallographic  b,  it  retains  its 
position  because  the  refraction  is  the  same  for  light  of  all  colors.  The  obtuse 
bisectrix,  and  0  and  7  or  a,  lie  in  the  symmetry  plane,  whereby,  in  this  case, 


FIG.  638.  FIG.  639.  FIG.  640. 

FIGS.  638  to  640. — Crossed  dispersion  of  the  obtuse  bisectrix. 

the  obtuse  bisectrix  and  the  optic  angle  are  dispersed.  This  dispersion 
was  called  dispersion  tournnatc  or  dispersion  croisee  by  Des  Cloizeaux,2  since 
by  successive  monochromatic  illuminations  in  the  order  of  the  spectrum, 
the  interference  figure  will  successively  occupy  the  positions  between  pp 
and  vv  in  Fig.  638,  and  will  appear  to  rotate  about  the  b  axis.  In  white 
light  the  distribution  of  color  in  the  two  melatopes  differs  from  right  to  left  and 
from  top  to  bottom  (Figs.  639-640).  The  colors  occur  in  inverse  order, 
so  that  if  the  figure  is  rotated  through  180°  about  the  acute  bisectrix,  each 
melatope  will  occupy  the  position  previously  held  by  the  other,  and  the  color 
distribution  will  be  exactly  the  same  as  before.  When  the  interference  figure 
is  in  parallel  position,  so  that  its  bars  coincide  with  the  principal  planes  of 
the  nicols,  the  black  cross  is  unsymmetrically  bordered  by  colored  bands 
(Fig.  639).  That  is,  the  upper  left  and  lower  right  sides  are  bordered  by  one 
color,  and  the  upper  right  and  lower  left  by  another. 

1  Op.  cit. 

2  Op.  cit. 


448  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  387 

PROBLEM 

Examine  a  crystal  of  borax  between  crossed  nicols. 

DISPERSION  IN  TRICLINIC  CRYSTALS 

387.  Unsymmetrical  Dispersion. — In  triclinic  crystals  the  shape  of  the 
optical  ellipsoid  (indicatrix)  as  well  as  its  position  in  the  crystal,  differs  for 
different  colors,  whereby  the  three  principal  vibration  axes  are  all  more  or 
less  dispersed.  An  interference  figure  from  such  a  section  of  such  a  crystal 
cut  at  right  angles  to  the  acute  bisectrix  will  show,  in  white  light,  a  totally 
unsymmetrical  figure  (Figs.  641-642). 


FIG.  641.  FIG.  642. 

FIGS.  641  and  642. — Unsymmetrical  dispersion  in  triclinic  crystals. 

388.  Effect  of  Temperature  Change  on  Dispersion. — The  indices  of  re- 
fraction of  transparent  substances  are  generally  decreased  by  an  increase  in 
temperature,1  therefore  the  axes  of  the  optical  ellipsoid  are  changed  and, 
consequently,  the  optic  axial  angle.  This  change  may  be  so  great  that  the 
axial  plane  for  one  temperature  may  lie  at  90°  to  that  at  another,  as  was 
first  observed  by  Brewster2  in  glauberite.  In  such  crystals,  at  some  inter- 
mediate temperature,  the  interference  figure  is  uniaxial;  for  example,  in 
gypsum  between  105°  and  115°,  the  value  differing  slightly  in  different 
crystals. 

1  A.  E.  H.  Tutton:  Crystallography  and  practical  crystal  measurement.     London,  1911, 
Chapter  XL VIII.     Also  plate  III,  p.  790. 

Idem:  Op.  cit.  supra,  in  Art.  382. 

E.  H.  Kraus  and  L.  J.  Youngs:  Ueber  die  Aenderungen  des  optischen  Achsenwinkels 
in  Gips  mit  der  Temperatur.  Neues  Jahrb.,  1912  (I),  123-146. 

Edward  H.  Kraus:  Die  Aenderungen  des  optischen  Axenwinkels  im  Glauberit  mit  der 
Temperatur.  Zeitschr.  f.  Kryst.,  LII  (1913),  321-326. 

2  Sir  David  Brewster:    On  the  action  of  heat  in  changing  the  number  and  nature  of  the 
optical  or  resultant  axes  of  glauberite.     Phil.  Mag.,  (3),  I  (1832),  417-420. 


CHAPTER  XXXII 

THE  PETROGRAPHIC  MICROSCOPE  AS  A  CONOSCOPE  AND  THE 
METHODS  FOR  OBSERVING  INTERFERENCE  FIGURES 

389.  Observing  Interference  Figures  with  the  Microscope. — A  general 
description  of  the  petrographic  microscope  as  a  conoscope  was  given  in 
Article  354,  and  it  was  there  mentioned  that  the  image  of  the  interference 
figure  is  formed  in  the  tube  too  far  below  the  ocular  to  be  seen  through  it 
without  the  insertion  of  an  accessory  lens  (Fig.   511).  '  The  fact  that  it 
could  there  be  seen  was  long  unrecognized,  for  although  interference  figures 
had  been  obtained  and  studied  in  large  crystal  sections  by  means  of  the 
Norrenberg  polarization  apparatus,  and  later  by   Groth's1   conoscope,   no 
method  for  their  observation  under  the  microscope  was  given  until  Lasaulx 
published  his  description  in  1878. 

390.  Lasaulx  Method  (1878). — Lasaulx2  recognized  the  fact  that  inter- 
ference figures  could  be  seen  by  simply  removing  the  ocular.     He  found 
that  a  high-power  objective  and  a  stronger  system  of  condensing  lenses  than 
up  to  that  time  had  been  placed  above  the  polarizer,  would  increase  the 
size  of  the  figures.     Such  condensing  lenses  are  now  universally  attached 
to  petrographic  microscopes  and  Lasaulx's  method  of  observing  interference 
figures  is  still  commonly  used.     It  is  simply  necessary  to  remove  the  ocular 
and  look  down  the  tube  at  the  mineral  between  crossed  nicols.     The  objec- 
tion to  this  method  of  observation  is  that  by  removing  the  ocular,  the  cross- 
hairs or  micrometer  scale  are  removed  also,  and  no  means  remains  for 
measuring  the  positions  of  the  points  of  emergence  of  the  optic  axes.     Various 
kinds  of  oculars  have  been  invented  to  overcome  this,   and  these  will  be 
described  in  the  chapter  dealing  with  the  measurement  of  the  optic  axial 
angle.     The  difficulty  with  most  of  these  devices  is  that  while  the  interfer- 
ence figure  is  magnified,  it  is  also  made  less  sharp. 

391.  Bertrand   Method    (1878). — Independently   of  Lasaulx,  Bertrand3 

1  P.  Groth:  Ueber  Apparate  und  Beobachtungsmethoden  fur  krystallographisch-optische 
Untersuchungen.     Pcgg.  Ann.  CXLIV  (1872),  footnote,  p.  38. 

2  A.  v.  Lasaulx:  Ueber  die  Ver-wendung  des  Mikroskopes  als  Polar  isationsinslrument  im 
convergenlen  Lichte  und  ein  neues  Mikroskop  zu  miner alogischen  Zwecken.     Neues  Jahrb., 
1878  (March  7),  377-380. 

3  Emile  Bertrand:   De  V application  du  microscope  a  I 'etude  de  la  mineralogi?.     Bull. 
Soc.  Min.  France,  I  (1878),  22-28,  93-96. 

Idem:  Sur  Vexamen    des   miner  aux   en  lumiere  polarisee   comer  genie.     Ibidem,  VIII 
(1885),  29-31. 

Idem:  Nouvelle  disposition  du  microscope  perrncttant  de  mesurer  Vccartcmenl  des  axes 
optiqnes  et  les  indices  de  refraction.     Ibidem,  VIII  (1885),  377-383. 

L.  Calderon:  Review  of  articles  in  Bull.  Soc.  Min.  France,  I  (1878),  22-28,  96-97,  in 
Zeitschr.  f.  Kryst.,  Ill  (1879),  642-644. 

29  449 


450  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  392 

in  the  same  year,  described  a  method  by  which  interference  figures  not  only 
could  be  seen,  but  could  be  measured.  So  far  back  as  1844,  Amici1  had  used 
an  auxiliary  lens  in  the  conoscope  for  observing  and  magnifying  the  inter- 
ference figures  of  mineral  sections.  The  same  principle  was  used  by  Bertrand 
in  the  polarizing  microscope.  He  placed,  beneath  the  ocular,  an  achromatic 
lens,  which,  with  the  eyepiece,  formed  a  weak  independent  microscope,  and 
was  so  arranged  that  it  could  be  focussed  upon  the  primary  interference 
figure  lying  within  the  tube  (Fig.  511).  The  image  appeared  enlarged,  and, 
since  the  ocular  was  used,  the  distance  between  the  points  of  emergence  of 
the  optic  axes  could  be  measured.  Above  the  polarizer  he  used  two  very 
short  focus  lenses  to  produce  strongly  convergent  light,  thus  enlarging  the 
figure. 

In  his  second  paper,  Bertrand2  improved  his  lens  by  fastening  it  in  a  slider 
so  that  it  could  be  inserted  or  removed  quickly  from  the  axis  of  the  micro- 
scope. While  the  interference  figures  produced  by  the  Bertrand  lens  are  not 
quite  so  sharp  as  those  seen  by  the  Lasaulx  method,  yet  the  convenience  of 
observation,  and  the  ease  of  measuring  are  of  such  great  advantage,  that  all 
modern  microscopes  are  fitted  with  them.  Such  a  lens  should  be  permanently 
attached  to  the  microscope  in  a  slider  similar  to  that  carrying  the  analyzer, 
whereby  it  may  be  quickly  inserted  or  removed.  It  should  be  so  arranged 
in  a  sliding  sleeve  that  its  distance  from  different  oculars  may  be  regulated 
to  bring  the  interference  figure  sharply  into  focus  without  the  necessity  of 
raising  the  ocular  in  the  tube,  an  improvement  due  to  the  suggestion  of 
Laspeyres.3 

The  Bertrand  lens  is  sometimes  called  the  Amici-Bertrand  lens,  incor- 
rectly, according  to  Klein,4  for  the  Amici  lens  had  been  in  use  for  thirty-four 
years  without  it  having  occurred  to  any  one  before  Bertrand,  to  apply  it  to 
the  polarizing  microscope. 

392.  Klein  Method  (1876)  1878. — In  1876,  according  to  Cohen,5  Klein 
had  discovered  a  similar  method,  although  he  did  not  publish  it  until  i878.6 

1  M.  Amici:  Note  sur  un  appareil  de  polarisation.     Ann.  d.  Chim.  et  Phys.,  XII  (1844)' 
114-120. 

Translation  of  same:  Ueber  einen  Polarisations-Apparat  wn  Urn.  Amici.  Pogg.  Ann., 
LXIV  (1845),  472-475- 

2  Emile  Bertrand:  Op  cit.,  I  (1878),  96-97. 

?H.  Laspeyres:  Miner  do  gische  Bemerkungen.  Interferenzbilder  des  Piemontit.  Zeit- 
schr.  f.  Kryst.,  IV  (1880),  460-464. 

4  C.  Klein:  Ueber  das  Arbeiten  mil  dem  in  ein  Polarisations  instrument  umgewandellen 
Polarisationsmikroskop  und  iiber  eine  dabei  in  Betracht  kommende  vereinjachte  Melhode  zur 
Bestimmung  des  Charakters  der  Doppelbrechung.  Sitzb.  Akad.  Wiss.,  Berlin,  1893  (I), 
footnote  3,  p.  224. 

6  E.  Cohen:  Zusammenstellung  petrographischer  Untersuchungsmethoden.  ist  ed., 
1884,  footnote  i,  page  12. 

6  C.  Klein:  Ueber  den  Feldspath  im  Basalt  torn  Hohen  Hagen  bei  Gottingen,  etc.  Nachr. 
Gesell.  Wiss.  Gottingen,  1878,  461.* 


ART.  394]     THE  PETROGRAPHIC  MICROSCOPE  AS  A  CONOSCOPE  451 

Instead  of  using  a  lens  within  the  tube,  he  found  that  the  interference  figure 
could  be  seen  by  holding  a  hand  lens  at  the  proper  focus  above  the  Ramsden 
disk  of  the  ocular,  or  even  by  simply  placing  the  eye  at  or  beyond  the  distance 
of  distinct  vision,  and  looking  down  upon  the  Ramsden  disk.  Later1  he 
mentioned  another  simple  method  by  means  of  which  the  interference  figure 
might  be  magnified,  namely,  by  placing  directly  on  the  regular  eyepiece 
another  Huyghens'  ocular.  By  this  method  the  cross-hairs  or  micrometer 
scale  remain  in  the  field,  and  the  interference  figure  may  be  measured.  The 
method  is  not  applicable  to  all  microscopes  since  the  focal  plane  of  the  upper 
ocular,  which  should  fall  exactly  in  the  Ramsden  disk,  does  not  always  do  so. 
If  it  falls  below,  the  upper  ocular  may  be  slightly  raised;  if  it  falls  above,  the 
ocular  cannot  be  used  unless  it  is  reversed,  whereby  the  cross-hairs  dis- 
appear from  view. 

393.  Laspeyres  Method   (1880). — Laspey res2  attached  his  Bertrand  lens 
to  the  bottom  of  the  inner  tube  which  regulates  the  tube  length.     By  this 
means  the  Betrand  lens  and  the  ocular  move  together.     This  is  practically 
the  arrangement  adopted  in  modern  petrographic  microscopes  with  the  excep- 
tion that  the  distance  between  the  ocular  and  the  Bertrand  lens  can  be 
regulated  to  permit  the  use  of  oculars  of  different  magnifications. 

Laspeyres  also  used  a  combination  of  the  Lasaulx  and  the  Bertrand 
methods,  removing  the  ocular  but  using  the  Bertrand  lens.  He  claimed  that 
this  arrangement  gave  him  a  large,  sharp,  and  highly  magnified  field,  although 
it  did  not  permit  the  use  of  cross-hairs.  He  obtained  different  magnifications 
by  raising  or  lowering  the  Bertrand  lens,  and  obtained  in  the  field,  with  a 
Hartnack  microscope,  both  melatopes  in  a  crystal  of  topaz  (2E=i  20°). 
This  combination  method  does  not  work  equally  well  with  every  microscope, 
the  effectiveness  depending  largely  upon  the  character  of  the  Bertrand  lens. 

In  the  same  article,  Laspeyres  compared  the  Lasaulx,  Bertrand,  and  Klein 
methods,  and  found  that  the  first  and  third  gave  the  sharpest  figures,  though 
small  in  size,  but  did  not  permit  the  use  of  cross-hairs;  the  Bertrand  method 
gave  greater  magnification  and  preserved  the  cross-hairs,  but  the  image  was 
diminished  in  brightness  and  sharpness. 

394.  Bertrand  Method  (1880). — Bertrand3  made  the  discovery,  in  1880, 
that  in  the  bubbles  which  are  found  so  frequently  in  the  balsam  between  the 
mineral  section  and  the  cover-glass,  one  may  see  an  interference  figure  of  the 
mineral  lying  below  it.     The  bubble  acts  as  a  lens  of  short  focal  length  and 
takes  the  place  of  the  objective,  while  the  ocular  and  objective  combined  act 
as  do  the  Bertrand  lens  and  the  ocular  in  the  usual  Bertrand  method. 

1  C.  Klein:     Ueber  das  Arbeiten,  etc.,  page  226,  Cit.  supra. 

2  H.  Laspeyres:    Mineralogische  Bemerkungen.     Interferenzbilder  des  Piemontit.     Zeit- 
schr.  f.  Kryst.,  IV   (1880),  460-461. 

3  Emile  Bertrand :    De  V application  du  microscope  d  V etude  de  la  miner alogie.     Bull. 
Soc.  Min.  France,  III  (1880),  93-96. 


452  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  395 

395.  Schroeder  Van  der  Kolk  Method  (1891). — The  method  just  men- 
tioned was  carried  a  step  farther  by  Schroeder  van  der  Kolk.1  He  placed 
a  drop  of  glycerine  upon  the  cover-glass  of  the  slide  to  be  examined,  and  stirred 
it  rapidly  with  a  thin  rod  so  that  it  became  filled  with  very  small  bubbles. 
Over  this  he  placed  a  cover-glass  and,  between  crossed  nicols  and  with  a 
medium  power  objective  (Zeiss  B,  or  Hartnack  4),  he  focussed  upon  the  bub- 
bles. The  tube  was  then  depressed  a  very  little,  since  the  bubbles  themselves 
act  as  biconcave  lenses,  and  there  appeared  in  each  bubble  a  fine  interference 
figure.  Though  the  images  are  small,  the  optical  character  can  be  determined 
readily.  Even  the  dispersion  in  a  flake  of  muscovite  can  be  seen  if  it  is  exam- 
ined without  a  cover-glass. 

This  method  of  observing  interference  figures  possesses,  in  certain  cases, 
very  great  advantages.  Since  no  change  is  made  in  the  arrangement  of  the 
microscope,  it  is  possible  to  observe  the  mineral  and  the  interference  figures 
at  the  same  time,  and  likewise  to  see,  simultaneously,  the  interference  figures 
of  a  number  of  grains,  whereby  their  optical  orientation  can  be  compared. 
The  method  is  especially  of  advantage  in  examining  fragments  which  are  too 
small  to  produce,  by  any  other  method,  even  with  the  use  of  a  diaphragm,  inter- 
ference figures  undisturbed  by  the  surrounding  grains.  Schroeder  van  der 
Kolk  found,  further,  that  mineral  fragments  must  be  somewhat  larger  than, 
the  bubble  to  show  the  figures  clearly,  and  that  the  farther  the  bubble  is 
removed  from  the  mineral,  the  larger  must  the  latter  be  to  give  a  good  figure. 
It  is,  therefore,  of  advantage,  for  very  small  grains,  to  remove  the  cover-glass 
of  the  rock-slice.  Bubbles  as  small  as  0.002  mm.  in  diameter  can  be 
used. 

To  avoid  the  necessity  each  time  of  preparing  anew  a  glycerine  foam 
Schroeder  v.  d.  Kolk  used  the  following  simple  device:  He  placed  a  drop  of 
Canada  balsam  on  an  object-glass  and,  if  necessary,  cooked  it.  After  pro- 
ducing foam  by  rapid  stirring,  he  placed  a  cover-glass  upon  it,  and  the  instru- 
ment was  complete.  To  use  it  he  placed  it,  cover-glass  downward,  upon  the 
rock-section,  and  shoved  it  into  such  positions  that  bubbles  appeared  over  the 
desired  spots. 

In  place  of  bubbles  small  drops  of  a  highly  refracting  fluid  can  be  used,  but 
in  that  case  the  tube  of  the  microscope  must  be  raised  instead  of  lowered  to 
get  the  image  into  focus.  To  make  these  bubbles  of  very  small  size  a  little 
oi  monobromnaphthaline  is  placed  on  an  object-glass.  Above  it,  at  a  distance 
of  i  cm.,  the  section  to  be  examined  is  held  with  the  cover-glass  down.  The 
a  monobromnaphthalene  is  heated,  and  the  distillation  product  condenses,  in 
fine  bubbles,  on  the  cover-glass  of  the  mineral  slice,  especially  if  it  is  cooled 
somewhat  by  placing  a  few  drops  of  water  on  the  top  surface. 

1  J.  L.  C.  Schroeder  van  der  Kolk:  Ueber  einc  Melhode  zur  Beobachtung  der  optischen 
Interferenzerscheinungen  im  convergenten  polarisirten  Lichte,  insbesondcre  in  Gesteinsschliffen. 
Zeitschr.  f.  wiss.  Mikrosk.,  VIII  (1891),  459-461. 


ART.  398]     THE  PETROGRAPHIC  MICROSCOPE  AS  A  COXOSCOPE 


453 


396.  Czapski  Ocular  (1893). — Czapski,1  in  1893,  made  an  improvement 
in  the  method  of  observing  interference  figures  by  the  Lasaulx  method,  by 
inserting  a  diaphragm  in  the  focal  plane  of  a  Ramsden  eyepiece  (Fig.  643).  A 
shoulder  holds  the  ocular  in  such  a  position  above  the  eyepiece  of  the  micro- 
scope, that  the  diaphragm  i  lies  in  the  plane  of  the  Ramsden  disk.  Upon 
removing  the  ocular  itself,  the  interference  figure  of  the  central  grain  only 
appears,  undisturbed  by  the  minerals  surrounding  it. 


Anat.Gr. 


FIG.    643. — Czapski     ocular. 
3/4  natural  size.      (Fuess.) 


FIG.    644. — Becke-Klein  magnifier. 
3/4  natural  size.     (Fuess.) 


397.  Becke-Klein  Magnifier  (1895). — Becke2  combined  Klein's  method 
of  observing  interference  figures  with  Czapski's,  by  which  means  sharp  and 
magnified  figures  may  be  obtained,  and  the  cross-hairs  or  micrometer  scale 
in  the  ocular  retained.     Above  a  Czapski  ocular  with  iris  diaphragm  (Fig. 
643)  is  placed  a  magnifying  glass  (O,  Fig.  644)  in  a  casing,  and  so  adjusted, 
that  when  placed  on  the  shoulder  of  the  Czapski  ocular,  the  diaphragm  of  the 
latter  appears  in  sharp  focus.     The  micrometer  scale  M  is  then  adjusted  by 
means  of  the  screws  k  and  k\,  until  it  lies,  without  parallax,  in  the  plane  of  the 
image.     To   this   apparatus   Becke   gave   the   name    of    Klein    magnifier 
(Klein'sche  Lupe)  since  Klein3  first  made  use  of  a  lens  above  the  Ramsden 
disk  for  examining  interference  figures. 

398.  Lenk-Lasaulx  Method  (1895). — To  preserve  the  sharpness  of  the 

1  S.  Czapski:     Ueber  Einrichtungen   behufs  schnellen   Ueberganges  vom  parallelen  sum 
comer  gcnten  Lichte  und  die  Beobachtung  der  Axenbildtr  von  sehr  kleinen  Krystallen  in  Polar- 
isations-Mikroskopen.     Zeitschr.  f.  Kryst.,  XXII  (1893),  158-162. 

C.  Leiss:  Die  optischen  Instruments,  etc.,  Leipzig,  1899,  218. 

2  F.  Becke:    Klein'sche  Lupe  mil  Mikrometer.     T.  M.  P.  M.,  XIV  (1894-95),  375-378. 
C.  Leiss:  Die  optiscJien  Instrumente,  etc.,  Leipzig,  1899,  218-219. 

3  See  Art.  392. 


454 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  399 


interference  figure  and  still  be  able  to  measure  it,  Lenk1  used  the  Lasaulx 
method  but  inserted  a  micrometer  scale  engraved  on  glass  between  the  lenses 
of  the  objective  (Fuess  No.  7)  and  in  its  upper  focal  plane  (or  above  the  upper 
lens  in  the  Zeiss  DD  objective).  The  divisions  of  the  micrometer  scale  may 
be  made  clearer,  without  reducing  the  sharpness  of  the  interference  figure, 
by  means  of  a  long  focus  lens  placed  in  the  tube  of  the  microscope. 

399.  Sommerfeldt  Condenser  (1895). — For  the  examination  of  mineral 
grains  which  are  so  minute  that  it  is  necessary  to  use  a  very  small  stop  to 
cut  out  interference  from  adjacent  minerals,  thereby  reducing  the  light  too 
much  to  give  a  figure  capable  of  examination  by  the  Bertrand  lens,  Sommer- 
feldt2 proposed  a  new  condenser  made  on  the  principle  of  that  of  ten  Siethoff.3 

i  On  the  flat  side  of  the  lower  lens  of  a  con- 
densing  system  of  three  plano-convex  lenses 
is  added  a  scale  whose  image  appears  in 
the  same  plane  as  the  interference  figure, 
as  seen  by  the  Lasaulx  method.  The 
optic  axial  angle  of  sections  cut  at  right 
angles  to  the  acute  bisectrix,  therefore,  may 
be  measured  by  this  means  in  the  same 
manner  as  though  the  Bertrand  lens  were 
used. 


FIG.  645. — Wright  reflecting  prism. 


400.  Wright-Lasaulx  Method  (1906). 
— To  avoid  the  necessity  of  removing  the 
ocular  for  the  observation  of  interference  figures  by  means  of  the  Lasaulx 
method,  Wright4  suggested  the  use  of  the  reflecting  apparatus  shown 
in  Fig.  645.  By  means  of  two  prisms  set  in  a  slider,  the  light  is  reflected  to 
the  side  of  the  tube  where  the  interference  figure  may  be  seen  without  the 
removal  of  the  ocular.  The  loss  of  light  by  reflection  is  not  great  enough 
appreciably  to  decrease  the  brilliancy  of  the  interference  figure.  A  specially 
ground  rectangular  prism  of  the  form  shown  at  the  right  was  suggested  by 
Wright5  as  an  improvement,  requiring  to  be  adjusted  but  once,  and  needing 
no  special  arrangement  of  bearings  for  the  prisms. 

401.  Johannsen  Auxiliary  Lens  (1912). — By  means  of  a  small  auxiliary 

*H.  Lenk:    Messung  des  Winkels  der  optischen  Axen  im  Mikroskope.     Zeitschr.  f. 
Kryst.,  XXV  (1895-96),  379~38o. 

C.  Leiss:  Einrichtung  zur  Axenwinke^messung  am  Mikroskop  nach  H.  Lenk.     Neues 
Jahrb.  B.  B.,  X  (1895-96),  429-430. 

2  E.  Sommerfeldt :    Die  mikroskopische  Achsenwinkelbestimmung  bei  sehr  kleinen  Kristall- 
praparaten.     Zeitschr.  f.  wiss.  Mikrosk.,  XXII  (1905),  356-362. 

3  Art.  254. 

4  Fred.  Eugene  Wright:  A  modification  of  the  Lasaulx  method  for  observing  interference 
figures  under  the  microscope.     Amer.  Jour.  Sci.,  XXII  (1906),  19-20. 

6  Idem:  In  litter  is,  Nov.  13,  1912. 


ART.  401]     THE  PETROGRAPHIC  MICROSCOPE  AS  A  CONOSCOPE  455 

lens  placed  directly  upon  the  cover-glass  of  the  mineral  under  examination, 
Johannsen1  was  able  to  obtain  interference  figures  of  extremely  small  grains, 
and  at  the  same  time  observe  the  mineral  by  parallel  light.  The  method  de- 
pends on  the  same  principle  as  does  that  of  Schroeder  van  der  Kolk,2  but 
the  lens  is  constructed  of  glass.  Johannsen  heated  a  small  glass  rod  in  the  flame 
of  a  Bunsen  burner,  and  drew  it  out  into  a  thin  thread,  which  was  again 
heated  and  again  drawn  out  to  hair-like  thinness.  After  breaking  the  glass 
into  pieces  3  or  4  cm.  in  length,  their  extremities  were  held  an  instant  in  the 
edge  of  the  flame,  whereby  truly  spherical  globules  were  produced  at  each 
end.  After  preparing  a  number  of  these  spherical  lenses,  they  were  examined 
under  the  microscope,  and  all  that  were  not  perfect,  or  which  contained  bub- 
bles, were  rejected.  Likewise  only  those  which  had  a  diameter  of  less  than 
o.  i  mm.  were  retained.  If  such  a  lens  is  placed  directly  in  contact  with  the 
cover-glass  of  the  mineral  to  be  examined,  and  the  microscope  arranged  with 
crossed  nicols,  ocular,  and  a  medium  or  low-power  objective  (No.  o  to  4,  Fuess) , 
there  will  appear  in  it  a  small  but  perfect  interference  figure.  The  micro- 
scope should  be  focussed  upon  the  glass  sphere,  after  which  the  tube  should  be 
slightly  raised.  A  condensing  lens  is  not  necessary,  but  without  it  part  of 
the  figure  is  cut  off  by  the  dark  border.  The  optical  character  and  dispersion 
can  be  determined  as  well  by  this  method  as  by  the  use  of  a  Bertrand  lens, 
and  the  figure  is  decidedly  sharper.  By  its  means  it  is  possible  to  examine 
the  interference  figures,  undisturbed  by  surrounding  minerals,  of  grains 
smaller  than  is  possible  by  the  Lasaulx,  Klein,  or  Bertrand  methods,  and  it 
possesses  the  further  advantage  that  the  mineral  and  the  interference  figure 
can  be  seen  at  the  same  time.  By  shifting  the  lens,  the  optical  orientation 
of  all  of  the  grains  in  a  section  can  be  determined.  When  used  with  a  von 
Fedorow  stage,  it  is  possible  to  examine  interference  figures  with  low-power 
objectives,  a  considerable  advantage  in  roughly  determining  the  orientation 
of  the  mineral  and,  consequently,  the  setting  of  the  stage. 

To  permit  the  rapid  examination  of  a  slide,  the  rod  to  which  the  glass 
globule  is  attached  may  be  fastened  with  a  bit  of  soft  modeling  wax  to  the 
edge  of  the  stage  at  such  an  angle  that  the  lens  rests  against  the  cover-glass 
of  the  mineral  section.  The  latter  may  now  be  shifted  around  the  stage  as 
much  as.  desired,  bringing,  successively,  the  different  mineral  constituents 
under  the  lens,  which  remains  undisturbed  in  the  center  of  the  field.  Another 
method  is  to  attach  the  rod  of  the  lens,  by  means  of  wax,  to  the  rim  of  a  cork 
or  wood  ring,  allowing  the  lens  to  project  toward  the  center,  with  the  rod  so 
tilted  that  the  lens  rests  on  the  same  plane  as  the  bottom  of  the  ring.  This 
method  protects  the  delicate  glass  rod  better  than  the  first,  but  is  not  quite 

1  Albert  Johannsen:    An  accessory  lens  for  observing  interference  figures  of  small  mineral 
grains.     Jour.  Geol.,  XXI  (1913)  96-98. 

2  Art.  395,  supra. 


456 


MANUAL  OF  PETROGRAPHIC  METHODS 


I  ART.  402 


so  convenient,  since  both  rock  section  and  cork  must  be  moved  when  it  is 
desired  to  place  different  minerals  in  the  center  of  the  field. 

402.  Orientation  of  Image  in  Relation  to  Object. — In  determining  the 
direction  of  inclination  of  the  optic  axes  or  the  bisectrix,  it  is  important  to 
know  the  orientation  of  the  interference  figure  in  relation  to  the  object  it- 
self or  to  its  image.  Klein1  tabulated  the  results  for  the  Lassaulx,  Klein, 
and  Bertrand  methods.  The  table  is  here  extended  to  include  several  others. 


Method 

Position  of  image 
in  relation  to 
object 

Position  of  inter- 
ference figure  in 
relation  to  object 

Position  of  inter- 
ference figure  in 
relation  to  image 

Microscope  as  ordinarily  used   . 

Inverted  .  . 

Lasaulx  method  

Parallel  ... 

Inverted 

Klein  method 

Inverted 

Parallel 

Bertrand  (1878)  .  .  .'  

Inverted 

Parallel 

Laspeyres  

Inverted 

Parallel 

Schroeder  v  d   Kolk 

Parallel 

Inverted 

Becke-Klein  .  .  . 

Inverted 

Parallel 

• 

Lenk-Lasaulx  

Parallel 

Inverted 

Sommerf  eldt-Lasaulx  

"Parallel  

Inverted. 

Wright-Lasaulx  

Parallel  .  .  . 

Inverted 

Johannsen  

Inverted 

Parallel 

1  C.  Klein:  Ueber  das  Arbeiten  mil  dem  in  ein  Polarisations  instrument  umgewandeUen 
Polar isationsmikroskop  und  iiber  eine  dabei  in  Betracht  kommende  vereinfachte  Methode  zur 
Bestimmung  des  Charakters  der  Doppelbrechung.  Sitzb.  Akad.  Wiss.  Berlin,  1893  (I),  227- 

E.  Weinschenk:  Review  of  above.     Zeitschr.  f.  Kryst.,  XXV  (1896),  607-609. 


CHAPTER  XXXIII 

DETERMINATION  OF  THE  OPTICAL  CHARACTER  OF  A  CRYSTAL 
BY  MEANS  OF  ITS  INTERFERENCE  FIGURE 

403.  Positive  and  Negative  Minerals. — Biot,1  in  1814,  discovered  that 
quartz  and  beryl  acted  differently,  in  polarized  light,  toward  minerals  whose 
action  was  known,  and  distinguished  them  as  attractive  and  repulsive.2  For 
these  terms  Brewster3  substituted  positive  and  negative;  terms  which  have  con- 
tinued in  use  to  the  present  time.  As  mentioned  in  Arts.  51  and  75,  uniaxial 
crystals  are  considered  positive  when  crystallographic  c  is  the  direction  of 
vibration  of  the  slow  ray,  and  biaxial  crystals  are  considered  positive  when 
the  slow  ray  ( c)  vibrates  parallel  to  the  acute  bisectrix. 

Various  accessories  may  be  used  in  making  the  determination  of  the  optical 
character  of  a  crystal,  the  most  common  being  the  mica  plate,  the  gypsum 
plate,  and  the  quartz  wedge.  The  history  of  the  introduction  of  these  acces- 
sories was  fully  given  above;4  here  their  application  to  the  determination  of 
the  optical  characters  of  crystals,  by  means  of  their  interference  figures,  will 
be  considered. 

UNIAXIAL  CRYSTALS 

404.  Quarter  Undulation  Mica  Plate. — Although  the  quarter  undulation 
mica  plate  was  introduced  as  a  compensator  by  Airy,5  in  1831,  it  was  first 
applied  to  the  determination  of  the  optical  character  of  crystals  by  Dove6 
in  1837. 

If  a  mica  plate  is  placed  above  a  mineral  giving  a  uniaxial  interference 
cross,  and  in  such  a  position  that  its  vibration  directions  lie  at  45°  to  those  of 
the  nicols,  it  will  be  found  that  the  center  of  the  cross  is  broken  apart  and 

1  J.  B.  Biot:  Memoire  sur  la  decouverte  d'une  propriete  nouvelle  dont  jouissent  les  forces 
polarisantes   de   certains  cristaux.     (Read  April  25,   1814.)     Mem.   Acad.  France,  XIII, 
Annee  1812,  pt.  ii,  Paris,  1814,  19-26. 

Idem:  Traite  de  physique.     Paris,  1816,  IV,  420-422,  543-566. 

2  Idem:  Addition  au  Memoire,  Sur  les  deux  genres  de  polarisation  exerces  par  les  cristaux 
doues  de  la  double  refraction.     (Read  May  15,  1814.)     Ibidem,  27-30,  especially  30. 

3  David  Brewster:  On  the  laws  of  polarization  and  double  refraction  in  regularly  crystal- 
lized bodies.     Phil.  Trans.  Roy.  Soc.  London,  CVIII  (1818),  199-273,  especially  219. 

4  Arts.  293,  294,  and  295. 

5  Art.  293. 

6  H.  W.  Dove:  Ueber  den  Unterschied  positiver  und   negativer  einaxiger  Krystalle  bei 
circularer  und  bei  elliptischer  Polarisation.     Pogg.  Ann.,  XL  (1837),  457-462.    . 

457 


458 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  404 


two  dark  spots  appear,  their  positions  depending  upon  the  optical  character 
of  the  mineral  under  examination.  The  phenomenon  observed  may  be  ex- 
plained as  follows: 

In  Fig.  646  the  two  rays  emerging  at  O  are  traveling  with  the  same 
velocity,  consequently  there  is  no  difference  of  phase  between  them.  This 
spot,  therefore,  will  appear  dark.1  At  a',  where  the  retardation  is  1/4  X, 
the  extraordinary  ray  will  be  vibrating  in  the  plane  Oa',  and  the  ordinary  ray, 
perpendicular  to  this  plane.  If  the  crystal  is  negative,  the  extraordinary  ray 
will  be  the  fast  ray  (e<  «,  E>0),  and  the  ordinary  ray  will  emerge  one-fourth 
of  a  wave  length  behind  it.  In  the  diagram,  the  ease  of  vibration  in  this  quad- 
rant may  be  represented  by  an  ellipse  in  which  the  diameters  represent 
both  the  vibration  directions  and  the  ease  of  vibration  of  the  fast  and  slow 
rays.  The  retardation  is  not  shown  in  the  figure.  At  another  point  a  of 
the  interference  figure,  the  extraordinary  ray  will  be  vibrating  in  the  plane 
Oa.  The  velocity  ellipse,  therefore,  in  this  quadrant  of  the  interference 


FIG.  646. 


FIG.  647. 


FIG.  648. 


FIGS.  646  to  648. — Positive  and  negative  uniaxial  interference  figures. 

figure,  will  lie  at  right  angles  to  that  in  the  northeast  quadrant.  In  the 
southwest  quadrant,  the  vibration  directions  are  parallel  to  those  in  the 
northeast,  and  those  in  the  northwest,  parallel  to  those  in  the  southeast. 

If,  now,  a  quarter  undulation  mica  plate,  with  its  slow  ray  vibrating  in 
a  direction  at  right  angles  to  its  long  edge  (Fig.  647),  is  placed  over  the  mineral 
producing  the  interference  figure,  the  retardation,  at  the  point  where  it  was 
originally  zero,  is  now  added  to  that  of  the  mica  plate,  and  the  sum  of  i/4X 
and  o  equals  1/4  X.2  At  a,  where  the  retardation  was  i/4X,  it  becomes  i/  2  X, 
since  the  vibration  directions  in  that  quadrant  are  parallel  to  those  in  the 
mica.  At  a',  however,  where  the  retardation  was  the  same,  it  now  becomes 
zero,  since  the  mica  plate,  also  1/4  X  but  with  vibrations  in  opposite  direc- 
tions, has  produced  exact  compensation  at  this  point.  In  a  similar  manner  b, 
originally  1/2  X,  becomes  3/4  X;  c  becomes  X;  and  so  on;  while  br,  originally 
i/  2  X,  becomes  1/4  X;  c'  becomes  3/4  X;  and  so  forth. 

1  Art  357. 

2  Art.  286. 


Fir,.  651.  PIG.  652. 

FIG.  649. — Interference  figure  of  calcite,  plate  cut  at  right  angles  to  the  optic  axis. 

FIG.  650. — Ditto,  combined  with  a  1/4  undulation  mica  plate  whose  slow  vibration  direction  lies 
X  E.-S.W.  Negative  character  of  the  calcite  shown  by  the  position  of  the  black  dots. 

FIG.  651.— Interference  figure  of  zircon  in  plate  cut  at  right  angles  to  the  optic    axis;   sodium  light. 

FIG.  652. — Ditto,  combined  with  a  mica  plate  whose  slow  vibration  direction  lies  N.E.-S.W 
Positive  character  of  the  zircon  shown.  (Facing  Page  458.) 


ART.  405]  THE  OPTICAL  CHARACTER  OF  A  CRYSTAL  459 

As  a  result  of  this  addition  and  subtraction  of  retardations,  the  originally 
symmetrical,  negative  interference  figure  (Fig.  649)  will  appear  as  shown  in 
Figs.  650  and  647,  with  two  dark  spots  near  the  center,  lying  along  the  di- 
rection of  the  slow  ray  of  the  quarter  undulation  plate. 

In  the  same  manner  it  may  be  shown  that  in  a  positive  uniaxial  crystal 
(Fig.  651),  the  two  dark  spots  will  lie  along  the  direction  of  the  fastest  ray 
of  the  mica  plate  (Figs.  648  and  652). 

The  above  description  applies,  of  course,  only  when  the  slow  vibration 
direction  of  the  mica  lies  at  right  angles  to  the  long  direction  of  the  plate, 
and  it  is  inserted  along  the  northwest -southeast  direction.  If  it  is  cut  with 
the  slow  ray  parallel  to  the  long  edge,  the  phenomenon  will  be  reversed. 

PROBLEMS 

Examine  the  interference  figure  of  calcite;  of  zircon.  Work  out  theoretically, 
in  the  same  manner  as  above,  the  location  of  the  black  spots,  using  a  mica  plate  cut 
with  the  slow  ray  parallel  to  the  long  direction. 

405.  Unit  Retardation  Plate. — When  a  gypsum  plate,  giving  red  or  violet 
of  the  first  order,  is  placed  over  a  uniaxial  interference  figure,  a  change,  sim- 
ilar  to  that  produced  by  the  mica  plate,  will  take 
place.     Instead   of  the  addition  or  subtraction  of  a          /<&r^\ 

/      /:•»::.       \ 

quarter  wave  length,  however,  there  is  now  a  change 
of  one  wave  length.  The  gypsum  plate  itself  gives  a 
red  or  violet  interference  color,  consequently  the  dark 
center  and  the  isogyres  of  the  interference  figure, 
having  no  influence  upon  the  overlying  plate,  be- 
come red.  In  a  positive  crystal,  and  with  a  gypsum 
plate  whose  slow  ray  vibrates  at  right  angles  to  the 
long  direction  of  the  plate,  the  position  correspond-  of  a  uniaxial  crystal,  cut  at 
ing  to  a',  Fig.  648,  originally  1/4  X  becomes  i  1/4  X,  right  "«le8  to  the ,  op'ic 

J      '  ^  '     axis.,     by     means     of     the 

and  a  becomes  37  4  X.  By  an  examination  of  Fig.  453  gypsum  plate.  Quartz  (+). 
or  the  tables  in  Arts.  276-277,  it  will  be  seen  that  an 

increase  of  a  quarter  of  a  wave  length  retardation  will  change  the  first  order 
red  to  blue,  and  a  decrease  of  a  quarter  wave  length  will  change  it  to  yellow. 
The  interference  figure,  shown  in  Fig.  653,  will,  therefore,  show  blue  spots  im- 
mediately adjacent  to  the  red  center  in  the  northeast  and  southwest  quadrants, 
and  yellow  spots  in  the  northwest  and  southeast.  The  phenomenon  of  color 
is  usually  much  more  pronounced  than  that  of  the  dark  spots  of  the  mica  plate, 
and  it  is,  therefore,  generally  advisable  to  use  the  gypsum  plate  for  inter- 
ference figures  produced  by  minerals  having  low  birefringence.  For  those 
having  high  birefrigence,  the  phenomena  produced  by  the  quartz  wedge  are 
most  easily  recognized. 

In  negative  uniaxial  crystals,  the  phenomenon  observed  is  reversed,  and 


460 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  406 


the  blue  spots  will  lie  in  the  northwest  and  southeast  quadrants.  If  the 
gypsum  plate  used  has  its  long  direction  parallel  to  the  slow  ray,  the  appear- 
ances seen  in  positive  and  negative  crystals  are,  of  course,  reversed. 

The  first  use  made  of  a  gypsum  plate  for  the  determination  of  the  optical 
character  of  interference  figures  was  by  Brewster,1  who  observed,  in  1818, 
the  different  colors  which  appeared  in 'the  alternate  quadrants.  No  definite 
thickness  of  plate  was  used,  however,  until  1835,  when  Brewster2  introduced 
the  II  order  red.  The  gypsum  plate  seems  to  have  fallen  into  disuse  and  was 
not  revived  until  1887,  when  a  I  order  red  was  among  the  accessories  used  by 
Rosenbusch,  although  he  did  not  publish  it  until  i892.3  It  was  discovered, 
independently,  by  Rhine4  in  1891. 

PROBLEM 

Examine  the  interference  figures  of  nephelite  and  quartz  with  a  unit  retardation 
plate. 


FIG.  654.      \  FIG.  655. 

FIGS.  654  and  655. — Determination  of  the  optical 
character  of  a  uniaxial  crystal,  cut  at  right  angles 
to  the  optic  axis,  by  means  of  the  quartz  wedge. 


406.  The  Quartz  or  Gypsum  Wedge. — Comparable  in  every  way  to  the 
phenomena  observed  with  the  mica  and  the  gypsum  plates,  are  those  seen 

with  a   quartz   or  gypsum  wedge. 

1 1  \  /^"  |P^\  /__  Instead,  however,  of  showing  a  sin- 
gle rise  of  color,  there  will  be  a  pro- 
gressive change,  due  to  the  increas- 
ing thickness  of  the  wedge  as  it  is 
pushed  forward.  As  a  result,  the 
interference  colors  will  appear  to 
move  toward  the  center  in  two 
opposite  quadrants,  and  away  from 

ft  jn  fa  others.       The  directions  of 
movement    may    be    Worked    OUt  in 

the  same  manner  as  was  done  for 

the  mica  plate.  If  the  wedge  has  its  slow  vibration  direction  perpendicular 
to  the  long  edge,  and  it  is  pushed  from  southeast  to  northwest,  with  its 
thin  end  foremost,  above  a  positive  mineral,  the  colors  will  appear  to  move 

away  from  the  center  in  the  northwest  and  southeast  quadrants,  and  to- 

^^ 

s~  l  David  Brewster:  On  the  laws  of  polarization  and  double  refraction  in  regularly  crystal- 
lized bodies.  Phil.  Trans.  Roy.  Soc.  London,  CVIII  (1818),  199-273,  in  particular  219-220. 

2  Idem:  Optics,  1835,  197.  * 

3  H.   Rosenbusch:    Mikroskopische  Physiographic  d.  Miner  alien.     Stuttgart,  3   Aufl., 
1892,  189-190. 

4  F.  Rinne:    Ueber  eine  einfache  Methodeden  Charakter  der  Doppelbrechung  im  conver- 
genten  polarisirten  Lichte  zu  bestimmen.     Neues  Jahrb.,  1891  (II),  21-27. 

Idem:  Notiz  liber  die  Bestimmung  des  Charakters  der  Doppelbrechung  im  convergenten 
polarisirten  Lichte  mit  Hulfe  des  Gypsbldttchen  vom  Roth  I  Ordnung.  Centralbl.  f.  Min., 
etc.,  1901,  653-655. 


ART.  407] 


THE  OPTICAL  CHARACTER  OF  A  CRYSTAL 


461 


ward  it  in  the  others  (Fig.  654).  If  the  crystal  is  negative,  the  reverse 
movement  takes  place  (Fig.  655).  With  a  wedge  cut  with  its  slow  ray 
parallel  to  the  long  direction,  the  phenomenon  for  the  positive  crystal  is  the 
same  as  given  for  the  negative  crystal  above,  and  vice  versa. 

The  quartz  wedge  was  introduced  by  Biot1  in  1814,  and  has  been  more  or 
less  used  ever  since. 

PROBLEM 

Examine  the  interference  figures  of  quartz  and  calcite  for  optical  character, 
using  the  quartz  wedge. 

Demonstrate  that  the  movement  takes  place  in  the  opposite  directions  from  that 
given  in  the  text,  when  the  long  direction  of  the  wedge  is  parallel  to  the  slow  ray. 

407.  Uniaxial  Crystals.  Inclined  Sections. — Inclined  sections  have 
exactly  the  same  effect  upon  the  accessories  as  do  sections  cut  at  right  angles 


TT& 


FIG.  656.  PIG.  657.  PIG.  658.  FIG.  659. 

FIGS.  656  and  657. — Determination  of  the  optical  character  of  a  uniaxial  crystal,  section  in- 
clined to  the  axis,  by  means  of  the  gypsum  plate. 

FIGS.  658  and  659. — Determination  of  the  optical  character  of  a  uniaxial  crystal,  section  in- 
clined to  the  axis,  by  means  of  the  quartz  wedge. 

to  crystallographic  c,  and  it  is  only  necessary  to  complete,  in  imagination,  the 
partial  interference  figure  seen  in  the  section.  Thus  Figs.  656  and  657  show, 
respectively,  the  northeast  and  the  southeast  quadrant  of  a  positive  uniaxial 
crystal  as  affected  by  a  gypsum  plate  (I  order  red).  With  the  quartz  wedge, 
the  movement  of  the  colors,  in  the  same  quadrants,  will  be  as  shown  in  Figs. 
658  and  659,  the  movement  being  much  clearer  in  the  southeast  quadrant, 
which  is,  consequently,  the  better  one  to  use  in  this  determination.  WTith 
negative  crystals,  or  with  the  fast  and  slow  directions  of  the  accessories 
reversed,  the  phenomena  are  reversed. 

PROBLEM 

Examine  inclined  sections  of  quartz  and  calcite  by  means  of  the  gypsum  plate 
and  quartz  wedge,  and  determine  their  optical  characters. 

1  J.  B.  Biot:  Memoire  sur  les  proprietes  physiques  que  les  molecules  lumineuses  acquir- 
ent  en  traversant  les  cristaux  doues  de  la  double  refraction.  Lu  22  Mai,  1814.  Mem.  Acad. 
France,  Annee  1812.  Paris,  1814,  31-38. 

Idem:  Traite  de  physique.     Paris,  1816,  IV,  420-422,  543-566. 


462 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  408 


408.  Sections  Parallel  to  the  Optic  Axis. — To  determine  the  optical  char- 
acter of  sections  cut  parallel  to  the  optic  axis,  the  position  of  this  axis  should 
first  be  determined  by  means  of  the  interference  colors,1  which  descend  in 
the  scale  outward  from  the  center  along  its  direction.  At  right  angles  to 
it  the  colors  rise.  Having  determined  the  direction  of  the  optic  axis 
(crystallographic  c),  the  optical  sign  of  the  elongation,  which  here  also  is 
the  sign  of  the  mineral,  may  be  determined  in  parallel  polarized  light. 


BIAXIAL  CRYSTALS 

409.  Mica  Plate,  Gypsum  Plate,  and  Quartz  Wedge. — Following  the  same 
method  of  reasoning  as  that  used  in  developing  the  phenomena  seen  in  uniaxial 
crystals,  we  may  determine  what  will  take  place  in  those  that  are  biaxial. 


PIG.  660. 


PIG.  66  r. 


PIGS.  660  and  66 1. — Vibration  directions  and  location  of  the  isogyres  in  biaxial  interference  figures. 

Let  Figs.  660  and  66 1  represent  the  interference  figures  of  a  biaxial  crystal, 
seen,  respectively,  in  parallel  and  in  diagonal  positions.     The  directions  of 


FIG.  662.  PIG.  663.  PIG.  664.  FIG.  665. 

FIGS.  662  to  665. — Movement  of  the  colors  upon  inserting  a  quartz  wedge  above  the  interference 
figures  of  positive  and  negative  minerals. 

vibration  and  transmission  of  the  rays  are  shown  as  developed  above.2     If 
the  crystal  is  negative,  the  acute  bisectrix  is  the  fast  ray  (a),  consequently 

1  Art.  359- 

2  Arts.  360  and  374. 


ART.  409] 


THE  OPTICAL  CHARACTER  OF  A  CRYSTAL 


463 


the  velocity  ellipses  of  rays  traveling  in  different  directions  appear  as  shown 
in  the  figures.  If,  now,  a  quarter  undulation  or  unit  retardation  plate,  or  a 
wedge,  is  inserted — vibration  directions  as  before — there  will  be  an  increase 
in  the  color  scale  in  that  part  of  the  figure  in  which  the  vibration  directions 
are  parallel,  and  a  decrease  where  they  are  at  right  angles.  The  resulting 
movement  is  exactly  the  same  as  that  which  takes  place  in  uniaxial  inter- 


PIG.  666.  FIG.  667. 

PIGS.  666  and  667. — Comparison  of  the  movement  of  the  colors  in  positive  uniaxial  and  biaxial 
minerals  upon  the  insertion  of  a  quartz  wedge  above  the  interference  figures. 

FIG.  668. — Biaxial  interference  figure  in  parallel  position,  combined  with  a  gypsum  plate. 


ference  figures,  as  may  be  seen  by  an  inspection  of  Figs.  662-665.  As  a 
matter  of  fact,  a  uniaxial  crystal  is  only  the  special  case  of  a  biaxial  crystal 
in  which  the  optic  angle  is  equal  to  zero,  and  the  two  may  be  considered  to- 
together.  This  is  clearly  brought  out  by  Figs.  666-667  which  show  the 
movement  produced,  by  a  quartz  wedge,  in  the  colors,  respectively,  of  a 
uniaxial  crystal  and  of  a  bi- 
axial crystal  placed  with  their 
vibration  directions  nearly  par- 
allel to  the  nicols.  Fig.  668 
shows  the  blue  and  yellow  spots 
produced  by  the  unit  retarda- 
tion plate1  in  a  biaxial  crystal 
similarly  placed. 

Becke2  showed  that  inclined 
sections  may  be  determined  in  FIG.  669.  FIG.  670. 

the  same  way  if  one  takes  note        FlGS-  669.  and  670.— Determination  of  the  optical 

.    .  character  of  biaxial  crystals,  in  sections  showing  the  emer- 

01    the  position   Of  the  aCUte  bl-     gence  of  one  optic  axis,  by   means  of  the  gypsum  plate. 

sectrix,  which   always   lies  on 

the  convex  side  of  the  hyperbola  when  the  crystal  is  rotated  to  its  diagonal 
position.  Thus  in  Fig.  669,  which  is  of  a  positive  mineral,  the  acute  bisectrix 
lies  to  the  northwest,  whereby,  if  the  gypsum  plate  is  used,  the  color  to  the 
southeast  of  the  red  will  be  yellow.  In  Fig.  670,  which  is  negative,  the  color 
southeast  of  the  red  will  be  blue. 


1  F.  Rinne:  Op.  dt. 

12  F.  Becke:  Die  Skiodromen.     T.  M.  P.  M.,  XXIV  (1905),  31-34. 


464 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  409 


With  a  wedge,  the  movement  in  inclined  sections  is  exactly  the  same  as  it 
is  in  uniaxial  crystals.  One  has  only  to  keep  in  mind  the  appearance  which  the 
interference  figure  would  have  if  it  were  complete.  Various  cases  are  illus- 
trated in  Figs.  671-673. 

If  one  neglects  determining  the  position  of  the  acute  bisectrix,  confusion 
will  result,  for,  except  for  the  curvature  of  the  isogyre,  the  left  melatope,  for 
example,  of  a  positive  mineral  (Fig.  671)  and  the  right  melatope  of  a  negative 
mineral  (Fig.  672)  are  alike,  and  wee  versa. 

When  the  point  of  emergence  of  the  optic  axis  lies  beyond  the  field  of  the 
microscope,  it  is  impossible  to  determine  the  curve  of  the  isogyre  when  the 
crystal  is  turned  to  the  diagonal  position  (Fig.  673).  It  is,  consequently, 
impossible  to  determine  the  location  of  the  acute  bisectrix  by  this  method. 


(t ) 


FIG.  671.  FIG.  672.  FIG.  673. 

FlGS.  671  to  673. — Movement  of  the  colors  in  the  interference  figures  of  biaxial  crystals  combined  with 
the  quartz  wedge.     The  section  is  inclined  to  the  optic  axis. 


Since  the  phenomena  seen  in  a  negative  mineral  showing  the  acute  bisectrix 
are  exactly  the  same  as  those  seen  in  a  positive  mineral  showing  the  obtuse 
bisectrix,  determinations  of  the  optical  character  in  such  sections  are  of  no 
value.  The  determination  whether  the  optic  angle  is  greater  or  less  than  90°, 
consequently  whether  the  acute  or  obtuse  bisectrix  lies  in  the  field,  will  be 
discussed  below.1  If  this  is  determinable,  the  optical  character  of  the  crystal 
can  be  determined  from  a  figure  inclined  as  much  as  that  shown  in  Fig.  673. 
Sections  of  biaxial  crystals  cut  parallel  to  the  plane  of  the  optic  axes  give 
interference  figures  as  shown  in  Fig.  552.  In  such  the  determination  of  the 
optical  character  is  easy,  as  was  shown  by  Becke.2  The  method  has  already 
been  given  in  reference  to  uniaxial  interference  figures.3  The  line  uniting 
the  quadrants  containing  the  lowest  colors  is  the  direction  of  the  acute  bi- 
sectrix, which  is  a  in  negative  and  c  in  positive  crystals.  When  the  axial 

1  Chapter  XXXIV,  infra.. 

2  F.  Becke:     Unterscheidung  von  optisch  +  und  —  zweiaxigen  Miner  alien  mil  dem  Mi- 
krokonoskop  (ah  Konoskop  gebranchtes  Mikroskop}.     T.  M.  P.  M.,  XVI  (1896),  181. 

Idem:  Die  Skiodromen.     T.  M.  P.  M.,  XXIV  (1905),  32. 

3  Arts.  359  and  408. 


ART.  409]  THE  OPTICAL  CHARACTER  OF  A  CRYSTAL  465 

angle  is   approximately   90°,    the  color  variation  becomes  indistinct,  when 
F  =  9o°,  it  disappears. 

PROBLEMS 

Examine  for  optical  character,  the  interference  figures  of  muscovite,  olivine, 
gypsum,  hornblende,  titanite. 

Examine  interference  figures  given  by  the  unit  retardation  plate,  or  by  cleavage 
flakes  along  the  best  cleavage  of  enstatite  (+),  and  hypersthene  (  — ). 


30 


CHAPTER  XXXIV 

MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  BY  CONVERGENT 

POLARIZED  LIGHT 

410.  Introductory.  —  In   a   previous   chapter1  the  relation  between  the 
optic  axes  and  the  axial  angle  was  discussed,  and  it  was  there  shown  that 

sin  E  =  n  sin  V, 

where  E  is  one-half  the  apparent  optic  angle  (Ao'B,  Fig.  674),  V  one-half  the 
true  optic  angle  (aob),  and  n  the  mean  refractive  index  of  the  crystal.  It  was 
also  shown  (Eq.  19,  Art.  71)  that 


where  Vf  is  one-half  the  axial  angle  whose  bisectrix  is  the  fast  ray,  and  a, 
/?,  and  7,  the  refractive  indices  of  the  substance  under  examination. 

To  measure  the  optic  axial  angle,  one  might 
pivot  the  crystal  at  O  and  rotate  it  until  the  line 
o'B  coincided  with  the  axis  of  the  microscope.  If 
a  reading  were  taken,  on  a  graduated  arc,  at  that 
position,  and  the  crystal  rotated  about  the  same 
axis  until  the  line  A  o'  coincided,  and  another  read- 
N  ing  taken,  the  resulting  angle  Ao'B  would  be  the 

FIG.   674.—  Relation   be-  apparent   axial   angle   (2E).     The  true  value  could 
tween  true  and  apparent  axial  tnen  be  determined   f  rom   the   first  equation  given 
above.     Petrographic  microscopes  are  not  ordinarily 

adapted  to  measuring  angles  of  rotation  in  the  plane  of  the  axis  of  the  micro- 
scope, although  special  apparatus  have  been  devised  for  this  purpose.2  In 
axial  angle  instruments,  the  method  of  rotation  is  the  one  usually  employed, 
but  such  measurements  belong  to  the  province  of  crystallography  rather 
than  to  petrography. 

Another  method  of  determining  the  axial  angle  follows  from  the  second 
equation,  and  it  is  clear  that  if  we  can  accurately  determine  the  values  of  the 
three  principal  indices  of  refraction,  we  can  compute  the  value  of  V.  For 
these  measurements,  also,  the  ordinary  petrographic  microscope  is  not  well 
adapted;  the  process  requiring  the  orientation  of  the  crystal  in  certain  defi- 
nite positions. 

1  Art.  72. 

2  Chapter  XXXV,  infra. 

466 


ART.  411] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


467 


The  usual  methods  for  the  determination  of  the  axial  angle,  by  means  of 
the  petrographic  microscope,  are  based  upon  an  examination  of  the  interfer- 
ence figures  produced  by  minerals.  The  results,  however,  cannot  be  quite 
so  exact  as  measurements  made  with  specially  designed  instruments,  owing 
to  the  fact  that  interference  figures  do  not  lie  in  a  plane  but  in  a  curved  sur- 
face, and  it  is  therefore  impossible  to  focus  sharply  upon  the  center  and  the 
edges  at  the  same  time.  Neither  is  it  possible  exactly  to  determine  the  points 
of  emergence  of  the  optic  axes,  since  they  are  represented  by  rather  broad  bars 
and  not  by  sharply  defined  points.  Nevertheless,  in  spite  of  these  drawbacks, 
the  measurements  made  by  the  microscope  in  convergent 
polarized  light  need  not  vary  more  than  a  few  degrees 
from  the  true  values,  provided  the  instrument  is  capable 
of  accurate  work  and  proper  precautions  are  taken  in 
making  the  determinations.1 

For  the  identification  of  minerals,  the  optic  angle 
may  be  measured  under  the  microscope  by  white  light. 
For  accurate  determinations,  however,  it  is  necessary 
to  use  monochromatic  light,  since  there  may  be  a  con- 
siderable difference  in  the  angle  for  light  of  different  wave 
lengths.  This  is  well  shown  by  the  interference  figures 
of  rubidium  platino  cyanide  (Figs.  632-637).  (See  also 
Figs.  522-523.) 

411.  Mallard  Method  for  Sections  showing  the 
Points  of  Emergence  of  Both  Optic  Axes  (1882). — The 
most  accurate  methods  for  determining  the  value  of  the 
axial  angle  under  the  microscope  are  based  upon  the 
work  of  Mallard,2  who  found  that  half  the  distance  be- 
tween the  points  of  emergence  of  the  optic  axes  (D,  Fig. 
675)  is  proportional  to  the  sine  of  half  the  angle  between 
them.  FlG-  6?5' — Diasram 

illustrating        Mallard's 

Let  parallel  rays  enter  the  crystal  section  (L,  Fig.    formula. 
675)  at  an  angle  of  v  with  the  normal.     The  rays  are  re- 
fracted when  they  pass  into  the  air  A ,  and  enter  the  lower  lens  O  of  the  ob- 
jective at  a  new  angle  u.     If  Of=f  =  the  focal  distance  of  the  objective,  and 
fg  =  d,  then 

d=fsinu.  (i) 

1  Cf.  S.  Czapski:  Die  dioptischen  Bedingungen  der  Messung  von  Axenwinkeln  mittelst 
des  Polarisationsmikroskops.     Xeues  Jahrb.,  B.B.,  VII  (1891),  506-515. 

2  Er.  Mallard:  Sur  la  mesure  de  I' angle  des  axes  optiques.     Bull.  Soc.  Min.  France,  V 
(1882),  77-87. 

See  also  B.  Hecht:  Ueber  die  Bestimmung  des  Winkels  der  optischen  Axen  an  Flatten 
der  en  Nor  male  nicht  mil  der  Halbirungslinie  des  Winkels  der  optischen  Axen  zusammcnfallt. 
Xeues  Jahrb.,  1887  (I),  250-261. 


468  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  412 

Since  the  relative  positions  of  the  objective,  Bertrand  lens,  and  ocular  remain 
constant,  the  distance  d  bears  almost  exactly  a  constant  ratio  to  the  corre- 
sponding distance  D,  seen  in  the  interference  figure,  therefore 


where  K  is  a  constant.     Substitute  this  value  in  equation  (i), 

—  =  f  sin  u. 
K 

But  u  may  be  replaced  by  E,  the  apparent  optic  angle  in  air,  whereby 

D  =  Kfsm  E. 

Since  K  and  /  are  constant  for  the  same  combination  of  lenses,  they  may 
be  replaced  by  K,  and  the  equation  becomes 


in  which  D  is  half  the  distance  between  the  points  of  emergence  of  the  optic 
axes,  K  a  constant  which  may  be  determined  for  any  combination  of  lenses, 
and  E  one-half  the  optic  axial  angle  in  air. 

Equation  (2)  is  known  as  Mallard's  formula.  The  constant  K,  known  as 
Mallard's  constant,  may  be  determined  very  simply  for  each  lens  combination 
of  a  given  microscope  by  using  as  a  test  plate  a  mineral  of  known  axial  angle,2 
measuring  the  distance  between  the  points  of  emergence  of  the  optic  axes 
(2D),  and  substituting  this  value  in  the  formula.  For  example,  using  a  flake 
of  muscovite  with  a  known  angle  of  2E  =  fji°  15',  the  distance  between  the 
melatopes  (2D),  with  a  Fuess  No.  7  objective  and  a  No.  2  ocular,  was  found 
to  be  29,  whereby 

K=.      I40'-5-    7  =  -I4^=24.0+. 
sm  35    37-5      0.5825 

If  possible,  determinations  should  be  made  on  a  number  of  minerals  with 
known  axial  angles.  The  mean  value  for  K  may  be  taken  as  a  reliable  value 
for  the  constant  for  that  particular  microscope  and  lens  combination. 

412.  Becke's  Graphical  Solution  of  sin  E  =  n  sin  V  (1894).  —  Instead  of 
calculating  the  value  of  2E  for  each  individual  case,  Becke3  constructed,  once 
for  all  for  a  certain  microscope  and  lens  combination,  curves  whose  abscissae 
represented  the  divisions  of  the  micrometer  scale  of  the  eyepiece,  and  whose 

1  This  formula  ha?  been  tested  by  several  writers  by  comparing  the  calculated  values 
with  those  obtained  by  experiment,  and  the  agreement,  in  general,  is  close  over  the  entire 
field.     See  Rosenbusch-Wiilfing:  Mikroskopische  Physiographic,  4  ed.,  1904,  330,  and  F.  E. 
Wright:  Measurement  of  the  optic  axial  angle,  Amer.  Jour.  Sci.,  XXIV  (1907),  327-331. 

2  Such  test  plates  may  be  obtained  from  most  dealers  in  petrographic  microscopes  and 
accessories. 

3  F.  Becke:  Klein'  sche  Lupe  mil  Mikromeler.     T.M.P.M.,  XIV  (1894),  375-378. 


ART.  413] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


469 


ordinates  represented  the  apparent  axial  angles  for  indices  of  1.5,  1.6,  and 
1.7.  A  specimen  of  a  diagram  of  such  curves  is  shown  in  Fig.  676,  plotted 
with  K  =  2$.  Different  curves  must  be  constructed,  of  course,  for  each  dif- 
ferent microscope  and  lens  combination  used.  From  such  a  diagram,  by 
interpolation,  the  true  axial  angle  for  any  refractive  index  may  be  determined. 
Another  diagram  to  express  the  relation  sin  E  =  n  sin  V,  is  shown,  as  con- 
structed by  von  Fedorow,1  in  Fig.  698. 


Graphical  solution  of  Mallard's  formula 
with  K  =  25.0 


160 
140 
120 
100 
80 


2D, 


FIG.  676. — Graphical  solution  of  Mallard's  formula. 


413.  Schwarzmann  Axial  Angle  Scale  (1896). — Another  method  for  de- 
termining the  values  of  2E  according  to  Mallard's  formula  is  by  means  of 
a  slide  rule,  upon  the  movable  part  of  wThich  the  position  for  each  lens  system 
may  be  marked.  Slide  rules,  however,  are  not  common  adjuncts  to  a  petro- 


n  TV  *•                                                                                                                  i          1        i              i      1 

Qll   4                                          •  —  !  !  ••  ;  '  r  !  —  •  L—  i  —  i  —  j  —  i  —  —  r-r—  ~ 

—  \J      ^  _!,  .  :  1  :  —  —  !  •  :  j-^  '  '  —  ;— 

jjjj    [T!  •        •  •  :  .  1  •.  ,  i  i  i  !  .•••.!::•''   ^!  :  '"  .     .        ~                   j 

^  ii*  A                                            1,1                                          20                       30               40          50        60             81)     '  100  '  120    180 

FIG.  677. — Schwartzmann's  axial  angle  scale.     (Fuess.) 

graphical  laboratory,  wherefore  Schwarzmann2  presented  a  scale,  based  on 
logarithmic  principles,  from  which  the  values  of  iE  may  be  read  directly. 

1  E.   von   Fedorow:    Universal    Methode   und  Feldspathstudien.     Zeitschr.    f.    Kryst., 
XXVI   (1896),   246  and  Fig.  3,  pi.  IV.     The  same  diagram,  drawn  to  a  larger  scale,  is 
given  by  Wright:  Methods,  etc.,  pi.  VII. 

2  Max  Schwarzmann:  Hilfsmittel  urn  die  Ausrechnung  der  Mallard 'schen  Formel  zu 
ersparen.     Neues  Jahrb.,  1896  (I),  52-56,  pi.  II. 

C.  Leiss:  Die  optischen  Instrumente  etc.,  Leipzig,  1899,  189-190. 


470  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  414 

The  scale  consists  of  two  parts,  of  which  one  (2!),  Fig.  677)lgiVes  microm- 
eter divisions  and  the  other  the  values  of  2E.  To  set  the  scale  for  any  par- 
ticular microscope  and  lens  combination,  the  number  of  divisions  (2!))  in 
the  micrometer  eyepiece,  corresponding  to  the  distance  between  the  mela- 
topes  in  a  mineral  whose  axial  angle  is  known,  is  determined.  (For  accurate 
measurements  the  determinations  should  be  made  by  monochromatic  light.) 
The  second  scale  is  placed  below  the  first  so  that  the  known  axial  angle  corre- 
sponds with  the  determined  number  of  micrometer  divisions  between  the 
two  melatopes.  As  a  check,  it  is  desirable  to  use  several  minerals  of  known 
optic  angles  before  determining  the  relative  positions  of  the  two  parts.  For 
convenience  in  use,  a  scale  may  be  prepared,  for  each  microscope  and  lens 
combination,  by  pasting  the  two  scales  in  proper  position  on  a  single  card. 

In  the  scale  shown  in  the  illustration,  for  example,  the  proper  setting  may 
have  been  obtained  by  noting  that  the  optic  angle  of  aragonite,  with  2E  = 
30°  15'  by  Na  light,  corresponded  to  17.7  divisions  of  the  micrometer,  and 
placing  the  two  scales  in  position  with  these  values  corresponding.  If,  now, 
barite  is  to  be  tested,  and  the  distance  between  the  hyperbolae  in  the  diagonal 
position  is  found  to  be  35.7  divisions,  the  value  of  2 E  is  63°  15'. 

The  Schwarzmann  scale  may  be  used,  further,  to  determine  the  value  of 
2  V  if  the  value  of  2 £  is  known.  Since  n  sin  V  =  sin  E,  log  sin  V  =  log  sin  E 
—log  n;  where  n  equals  the  mean  refractive  index  (/?)  of  the  mineral.  It 
is  therefore  only  necessary  to  lay  off  to  the  left,  from  the  mark  for  2E,  the 
distance  between  i  and  n.  For  example,  the  mean  refractive  index  of  barite 
is  1.638.  The  distance  between  1.638  and  i.o  is  determined  from  the  upper 
scale  and  is  laid  off  to  the  left  from  63°  15'  ( =  2E),  and  the  value  37°  (  =  27) 
is  obtained.  If  the  scales  are  arranged  to  slide  as  an  ordinary  slide  rule,  it 
would  be  necessary  simply  to  place  63°  15'  below  1.638  and  to  read  the  angle 
beneath  the  value  i.o. 

414.  Schwarzmann  Ocular  (1896). — Schwarzmann2 further  suggested  the 
convenience  of  having  an  ocular  directly  graduated  to  values  of  2E  instead  of 
the  usual  uniform  divisions,  the  values  being  those  of  sin  E  but  marked  from 
either  side  of  the  o  point  with  the  values  of  2  E.  Such  an  ocular  naturally  could 
be  used  only  with  one  particular  microscope  and  lens  combination.  If  the 
acute  bisectrix  did  not  fall  exactly  at  the  zero  point,  but  one  side  had  a  value 
of  2Ef  and  the  other  2E'+d,  the  value  of  2E  would  be  approximately 

2E'-\ .     For  example,  if  a  piece  of  barite  with  2^  =  63°  were  not  truly  cen- 
tered, so  that  one  side  read  58°  and  the  other  68°,  we  would  have 


=  63 


1  In  the  original  article  the  distance  between  the  two  eyes  is  given  as  D.      iD  is  here 
used  to  correspond  with  the  method  of  counting  the  divisions  in  the  Mallard  formula. 

2  Max  Schwarzmann :    Op.  cit.,  55-56. 


ART.  415] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


471 


415.  De  Souza-Brandao  Axial  Angle  Diagram  (1903). — Another  type  of 
protractor  for  determining  values  of  2E  from  Mallard's  formula  was  devised 


FIG.  678. — De  Souza-BrandSo  axial  angle  diagram.     (Fuess.) 


by  de  Souza-Brandao.1  It  consists  of 
a  rectangular  diagram  (Fig.  678),  15  by 
15  cm.,  upon  which  the  ordinates  rep- 
resent D  in  the  formula  D  =  K  sin  E,  and 
the  angular  graduations  represent  E. 
To  prepare  the  protractor  for  use,  a 
quadrant  of  a  circle  aa  is  drawn  with 
the  lower  left  corner  as  a  center  and 
with  the  observed  value  of  K  as  a  radius. 
(In  the  illustration  .£  =  3.225.)  The 
instrument  is  now  ready  for  use.  To 
determine  any  axial  angle,  a  ruler,  pref- 
erably one  of  celluloid  with  a  central 
mark  as  shown  in  the  figure,  is  extended 
from  the  lower  left  corner  through  the 
point  where  the  value  for  half  the  dis- 
tance between  the  melatopes  (in  this 

1  V.  de  Souza-Brandao:  0  novo  microscopic  da  commissSo  do  serviqo  geologi^o.  Com- 
municacoes  da  Commissao  do  Service  Geologico  de  Portugal,  V  (1903-1904),  118-250, 
in  particular  197-199, 


FIG.  679. — Trigonometer. 
tific  Co.) 


(Central  Scien- 


472  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  416 

case  2.160)  cuts  the  circle.  The  extension  of  this  line  to  the  protractor  edge 
gives  the  value  for  E.  In  the  figure,  £  =  42°,  whereby  the  apparent  optic 
axial  angle  in  air  (2£)  is  84°.  This  diagram  possesses  the  advantage  that 
any  number  of  circles  may  be  drawn,  each  representing  a  different  micro- 
scop  e  or  lens  combination. 

The  trigonometer  shown  in  Fig.  679  may  be  used  for  the  same  purpose. 

416.  Michel-Levy  Method  for  Sections  Perpendicular  to  a  Bisectrix 

(1888). — In  a  section  cut  at  right  angles  to  one  of  the  bisectrices  of  a  mineral 
whose  optic  angle  is  so  large  that  the  points  of  emergence  of  the  optic  axes 
lie  beyond  the  field  of  view  (2E>  ±85°),  it  is  impossible,  by  inspection,  to  de- 
termine whether  the  acute  or  the  obtuse  bisectrix  appears.  To  make  the 
determination,  Michel-Levy1  devised  a  method  by  means  of  which  it  is  pos- 
sible to  obtain  a  fairly  accurate  value  for  2E. 

A  glass  plate,  with  a  few  concentric  circles  engraved  upon  it,  is  inserted 
in  the  tube  of  the  microscope  between  the  analyzer  and  the  objective  and  in 
such  a  position  that  the  lines  appear  in  the  plane  of  the  interference  figure. 
The  section,  whose  angle  is  to  be  determined,  is  now  placed  on  the  stage  in 
parallel  position,  and  the  amount  of  rotation  necessary  to  bring  the  isogyres 
from  the  form  of  a  black  cross  to  the  point  of  tangency  to  a  given  circle,  is 
determined.  The  values  thus  obtained  are  substituted  in  the  formula 

.     ~     sin  £ 
sm  O  = 

in  which  E  is  half  the  apparent  axial  angle  of  a  known  mineral,  used  as  a 
measure  of  the  circle  of  reference,  n  the  refractive  index  of  the  glass  of  the 
objective,  and  <p  the  amount  of  rotation  of  the  stage  necessary  to  bring  the 
isogyres  of  the  unknown  mineral  from  the  crossed  position  to  the  point  of 
.  tangency. 

Since  the  value  of  n  is  usually  unknown,  Wright2  devised  a  formula  in 
which  Mallard's  constant  is  introduced  in  its  place.  This  formula  is  based 
on  the  equation  for  the  hyperbolic  isogyres  of  a  biaxial  interference  figure 

ocy  =  x'yf.      (Eq.  i,  Art.  379.)  (i) 

In  the  45°  position,  x  =  y,  and  the  equation  becomes 

x*  =  x'yr.  (2) 

In  polar  coordinates  x  =  p  cos  <p,  y  =  p  sin  <p,  x'  =  r  cos  <p,  and  y'  =  r  sin  <p, 
whereby  (2)  becomes 

p2  cos2  (p  =  r2  cos  <p  sin  <p.  (3) 

But  2  cos  (f>  sin  <p=  sin  2<p  and  (3)  becomes 

1  Levy  et  Lacroix:  Les  mineraux  des  roches.     Paris,  1888,  94-95. 

2  Fred.  Eugene  Wright:    The  determination  of  the  optical  character  of  birefr acting  min- 
erals.    Amer.  Jour.  Sci.,  XX  (1905),  288-9. 


ART.  4ir>]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 

2  p2  cosV  =r2  sin  2<p. 
When  .r  =  v,  cos2  p  =  (i/2\/2)2  =  1/2,  and  the  equation  reads 


473 


(4) 


Substituting  r  =  Z>,  in  Mallard's  formula,  we  have 

r  =  #sin  E. 


10  13  20         25      30        45 

FIG.  680. — Diagram  for  solving  the  equation  sin  E  =     /  .  (After  Wright.) 

Vsm  2<£ 

Likewise  in  the  circle  used  as  a  standard  of  measurement 

p  =  K  sin  O. 


Substituting  in  (4) 


ur 


sin  E 


--K*  snv  h-  sin  : 
sin  O  C 


Vsin  2^7     Vsin  2<p 
where  sin  O  (  =  C)  is  the  constant  of  the  circle  used,  and  is  to  be  determined, 


474 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  417 


once  for  all,  by  the  Mallard  method,  by  making  use  of  a  mineral  of  known 
axial  angle,  <p  is  the  angle  of  revolution  of  the  stage,  and  E  is  one-half  the 
apparent  axial  angle  of  the  mineral  to  be  determined. 

If  the  circle  of  reference  is  taken  so  that  its  diameter  is  equal  to  the  dis- 
tance between  the  melatopes  in  a  mineral  whose  value  of  2E  is  known,  it 
must  be  tangent  to  the  circle  in  the  diagonal  position  when  ^  =  45°-  In  this 
case  \/sin  2<p  =  i,  and  sin  E  of  the  known  mineral  =  sin  O  =  C,  a  constant. 

The  equation  does  not  hold  when  sin  2<p<  sin2  O,  for  in  that  case  sin  E 
in  equation  (5)  will  be  greater  than  unity.  This  is  due  to  the  fact  that  below 
this  point  the  apex  of  the  hyperbolic  isogyre  and  the  circle  of  reference  can- 
not be  tangent.  In  such  cases  it  is  necessary  to  choose  a  smaller  circle  of 
reference. 

The  value  of  E  can  readily  be  found  by  means  of  the  graphical  solution 
of  the  equation  shown  in  Fig.  680.  The  line  drawn  from  the  center  O  through 
the  intersection  of  whichever  circle  represents  the  angle  of  rotation  <p,  with 
that  horizontal  line  which  represents  the  constant  sin  0,  will  give  the  value 
of  E. 

417.  Viola  Method  (1893). — Viola1  determined  the  value  of  the  axial 
angle  in  sections  whose  optic  axes  emerge  beyond  the  field  of  view,  by  meas- 
uring the  curvature  of  the  hyperbola  in  any  position  of  the  stage. 


FIG.  681. 


FIG.  682. 
FIGS.  681-682. — Viola's  method  for  determining  the  axial  angle. 

The  accessories  necessary  are  a  Klein  ocular,  or  simply  two  oculars  used 
according  to  the  Klein  method,2  and  a  thin  glass  plate  engraved  with  con- 

1  C.  Viola:  Giornale  di  Miner,  etc.,  IV  (1893),  pt.  3.* 

Idem:  Ueber  den  Albit  von  Lakous  (Insel  Kreta),  T.M.P.M.,  XV   (1896),   135-158, 
especially  150-156. 

2  Art.  392. 


ART.  417] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


475 


centric  circles  and  graduated  with  divisions  5°  apart.  This  engraved  disk 
is  placed  in  the  focal  plane  of  the  upper  ocular  in  such  a  position  that  its 
center  does  not  correspond  with  that  of  the  ocular  (Fig.  681).  If  the  stage  is 
rotated,  the  successive  positions  assumed  by  the  isogyres  of  a  biaxial  mineral 
are  as  shown  in  Fig.  68 1.  If  the  stage  is  stationary  and  the  nicols  are  rotated, 
they  will  appear  as  in  Fig.  682.  The  latter  positions  are  the  ones  to  be  used. 
If  one  uses  a  microscope  whose  nicols  do  not  rotate,  it  is  necessary  to  compute 
the  positions  which  the  isogyres  would  have  assumed  had  they  done  so.  To 
do  this  one  must  lay  off,  in  the  opposite  direction  from  that  in  which  the  stage 
was  rotated,  the  angle  NCNz,  the  amount  of  the  rotation. 

To  determine  the  axial  angle,  the  stage  is  rotated  until  the  hyperbola 
passes  through  the  c'enter  of  the  graduated  disk.  The  angles  71  and  72  (Fig. 
683),  measured  from  the  left  and  clock- wise  to  the  points  where  the  hyperbola 
cuts  a  circle  of  radius  p,  are  read.  The  angle  co  (NZCN)  through  which  the 
stage  has  been  turned  from  the  position  of  forming  a  cross  to  its  position 
through  the  center  of  the  auxiliary  circle,  is  also  determined.  If  R  is  the 
distance  of  the  center  of  the  circle  from  the  center  of  the  hyperbola,  we  have 

R         sin  2  71  sin  272  ,  v 


2  sn 


sin  (7 


where 
and 


tan  i/  = 


71  tan  72 


AC     sin  2  \f/ 

R  ~sin  2  a)' 
K 


(3) 


PlG.  683. — Viola's  method  for  determining  the  axial  angle. 

The  value  of  p  is  determined,  in  degrees,  by  means  of  a  mineral  of  known 
optic  angle,  and  such  an  engraved  circle  is  chosen  that  its  diameter  corre- 
sponds with  the  value  of  2E(  =  2p)  of  that  mineral.  71  and  72  are  determined 
by  the  position  of  the  hyperbola  across  the  auxiliary  circle.  AC  =  E  =  one-half 
the  apparent  axial  angle,  co  is  determined  by  the  amount  of  rotation  of  the 
stage.  \f/  is  determined  from  equation  (2),  and  its  value  is  substituted  in 


476  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  418 

equation  (i)  to  determine  R  (in  degrees).  The  latter  value  is  substituted  in 
(3)  to  give  E. 

BECKE  METHOD  FOR  DETERMINING  GRAPHICALLY  THE   AXIAL   ANGLE  IN 

SECTIONS  WHICH  DO  NOT  SHOW  THE  POINTS  OF  EMERGENCE  OF 

BOTH  OPTIC  AXES  (1894) 

418.  Determination  of  the   Point  of  Emergence  of  an    Optic  Axis. — 

Becke's1  method  for  the  determination  of  axial  angles  was  developed  from 
a  method  for  determining  the  point  of  emergence  of  an  optic  axis.  The  meas- 
urement of  the  angle  will  be  more  readily  understood  if  this  is  previously 
explained. 

If  one  rotates  the  two  nicols  simultaneously  and  leaves  the  stage  station- 
ary, the  only  spot  which  will  remain  continually  dark  is  the  point  of  emergence 
of  an  optic  axis.2  If  the  stage  is  rotated  instead  of  the  nicols,  the  point  of 
emergence  of  the  optic  axis  will  rotate  in  the  same  direction,  following  the 
path  of  the  former  but  without  movement  of  its  own  (Fig.  68 1).  If,  now, 
one  can  accurately  locate  the  isogyre  in  two  positions,  their  intersection  will 
give  the  point  of  emergence  of  the  optic  axis. 

To  fix  the  position  of  the  isogyre  correctly  in  any  position,  it  is  sufficient 
to  determine  the  distance  from  its  nearest  point  to  the  center  of  the  field 
of  view,  and  the  two  points)  where  it  touches  the  periphery.  Two  positions  of 
the  isogyre  will  locate  the  optic  axis,  but  a  third  is  used  as  a  check.  The  first 
position  taken  is  one  in  which  some  crystallographic  line  lies  parallel  to  a 
cross-hair.  The  second  position  is  so  chosen  that  the  new  isogyre  crosses 
the  position  of  the  first  approximately  at  right  angles,  and  is  deter- 
mined by  rotating  the  stage  through  45°.  The  third  is  the  position  in 
which  the  isogyre  lies  parallel  to  one  of  the  cross-hairs.  In  the  process, 
the  direction  of  rotation  of  the  stage  must  be  taken  into  consideration, 
otherwise  confusion  will  result.  The  lines  representing  the  three  positions  of 
the  isogyre  should  intersect  in  a  point,  although  as  a  matter  of  fact  they  usu- 
ally form  a  small  triangle,  whose  size  is  an  indication  of  the  accuracy  of  the 
measurements.  The  center  of  the  triangle  may  be  taken  as  the  desired  point. 

An  example  will  best  illustrate  the  procedure.  Since  few  microscopes  are 
fitted  with  simultaneously  rotating  nicols,  the  appearances  as  seen  with  a 
rotating  stage  are  given.  Let  Fig.  684  represent  the  interference  figure  in  a 
section  of  plagioclase  when  it  is  placed  with  its  twinning  lamellae  parallel  to 
the  vertical  cross-hair  (zzf)  of  the  microscope,  and  let  this  be  the  first  position 
to  be  measured.  Let  it  be  supposed  that  the  vernier  of  the  stage  in  this 
position  reads  128°.  The  ocular  is  now  turned  until  the  vertical  cross-hair 

1  F.   Becke:    Bestimmung  kalkreicher  Plagioklase  durch  die  Interferenzbilder  von  Zwil- 
lingen     T.M.P.M.,  XIV  (1894-5),  4i5~442. 
2Cf.  Fig.  682,  Art.  417. 


ART.  418] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


477 


cuts  the  center  of  the  dark  bar  where  it  leaves  the  field  at  u.  The  Bertrand 
lens1  is  now  removed  and  the  stage  rotated,  clock-wise,  until  the  twinning 
line  (zz')  coincides  with  the  cross-hair  in  its  new  position  at  u.  The  stage 


FIG.  685. 


vernier  reads,  let  us  say,  154.5°  (  =  «).  Turn  back  the  stage  to  zz'  and  set 
the  vertical  cross-hair  of  the  ocular  on  r,  the  point  where  the  bar  leaves  the 
field  in  the  other  direction.  Upon  rotating  the  stage,  counter  clock-wise, 


z 

FIG.  686.  FIG.  687. 

FIGS.  684  to  687. — (After  Becke.) 

it  may  read  r  =  42°(  — ).  Turn  back  to  128°  and  set  the  ocular  so  that  its 
scale  extends  from  northwest  to  southeast  and  read  the  distance  between  the 
intersection  of  the  cross-hairs  and  the  bar,  for  example  ^  =  3. 

On  account  of  unavoidable  eccentricity  in  centering,  the  observations  are 
now  repeated  with  the  crystal  rotated  through  180°,  whereby,  in  the  above 
example,  zz;  =  i28°H-i8o°  =  3o8°,  the  new  reading  of  the  stage  vernier.  The 
other  values  may  be  #  =  353. 5°,  r  =  2i6°(  — ),  d=i.j.  Subtracting  180°  from 

1  If  a  Klein  ocular  is  used,  the  insertion  of  the  Bertrand  lens  permits  a  much  reduced 
image  of  the  mineral  to  be  seen. 


478  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  419 

each  of  the  first  three  readings  to  reduce  them  to  the  first  position,  we  have, 
upon  tabulation, 

z  u  r  d 

Treading 128°  154.5°  (+)     42°  (-)  3-0 

II  reading 128°  173-5°  36°  1.7 

Averages 128°  164°  39°  2.35 

Whereby  the  distances  ^-2  =  36°  (+),  ^-2  =  89°  (-). 

A  circle  is  now  drawn  with  a  diameter  equal  to  that  shown  by  the  mi- 
crometer divisions  of  the  eyepiece,  for  example,  13.0,  and  within  it  a  small 
circle  with  a  radius  of  2.35.  The  line  zz'  is  drawn,  the  points  u  and  r  are 
found,  and  a  curve  is  sketched  through  the  three  points  as  shown  in  Fig.  684. 

If  the  stage  is  now  rotated  approximately  45°  (in  the  example  43°)  counter 
clock- wise,  the  isogyre  appears  as  is  shown  in  Fig.  685.  The  position  of  the 
twinning  line  zz'  is  determined  in  its  new  position,  and  the  above  process  is 
repeated,  the  following  readings  resulting  after  deducting  180°  from  the 
second  set. 

I  position 85°  4i°(~)         99-5°(+)     3-3 

II  position 85°  32°  101.5°  2.6 

Averages 85°  36.5°  100.5°  2.95 

Whereby  r-z  =  48. 5°  (-),  and  o-z  =  15.5°  (+). 

The  third  reading  is  taken  by  rotating  the  stage  until  the  isogyre  lies  paral- 
lel to  one  of  the  cross-hairs,  which  in  this  particular  case  happened  to  be 
when  it  passed  through  the  center  of  the  field  (Fig.  686).  The  readings,  after 
reducing  the  second  set  by  180°,  are, 

z  a  d 

I  position 110°  36°  (  — )  o 

II  position 110°  37°  o 

Averages 110°  36.5°  o 

Whereby  a-z  =  73.5°  (-). 

If  the  three  bars  are  drawn  in  a  single  circle,  and  so  placed  that  the  line  zz' 
of  each  coincides,  they  will  appear  as  in  Fig.  687,  which  is  identically  what 
would  have  been  seen  by  using  a  microscope  with  simultaneously  rotating 
nicols  and  stationary  stage.  The  point  of  intersection  lies  73.5°  (  — )from 
zz'  and  3.6  micrometer  divisions  from  the  center,  the  latter  corresponding  to 
an  apparent  angle  of  27°. 

419.  Becke's  Rotating  Drawing  Stage  (1895). — Becke1  later  simplified 
the  process  of  construction  by  a  device  which  permits  a  complete  graphical 
solution.  Instead  of  measuring  the  various  positions  of  the  isogyre,  a  specially 
constructed  drawing-board  is  used  (Fig.  688),  and  the  image  is  drawn  by  means 

1  F.  Becke:  Messung  von  Axenbildern  mil  dem  Mikroskop.  T.M.P.M.,  XIV  (1894-5), 
563-656. 


ART.  419]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


479 


of  a  camera  lucida  (Z)  so  set  that  the  reflected  ray  falls  vertically  upon  the 
center  of  the  drawing-table.  The  latter  is  made  similar  to  the  rotating 
stage  of  the  microscope,  and  is  attached  by  three  centering  screws,  a,  a\,  and 
02,  to  the  support.  It  carries  a  vernier  at  one  side,  and  may  be  rotated  as 
desired.  The  microscope  is  placed  between  guide  plates  on  the  drawing- 
board  so  that  it  may  always  be  returned  to  the  same  position.  The  drawing 
paper  is  attached  to  the  rotating  stage  by  means  of  spring  object  clips. 


FIG.  688. — Becke  drawing  table.     About  1/5  natural  size.      (Fuess.) 

To  use  the  instrument,  the  microscope  is  first  centered  on  the  thin  section* 
the  camera  lucida  is  put  in  place,  the  ocular  is  removed,  and  the  image  of  the 
center  of  the  drawing-table  is  made  to  coincide  with  that  of  the  center  of  the 
microscope  stage.  After  obtaining  a  distinct  image  of  the  interference  figure, 
the  stage  vernier  is  read,  the  drawing  stage  is  set  to  the  same  reading,  and 
the  bar  sketched  in  this  position.1  The  microscope-stage  and  the  drawing- 
stage  are  now  both  rotated  through  approximately  30°  in  the  same  direction, 
and  again  the  black  bar  is  drawn.  The  intersection  of  the  two  gives  the  posi- 

1  Becke  suggests  using  black  drawing  paper  and  light  colored  crayons  for  sketching 
the  isogyres. 


480  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  420 

tion  of  the  optic  axis,  although  for  a  check  a  third  drawing  is  made  with  the 
stage  rotated  30°  in  the  opposite  direction.  The  position  of  twinning  or 
cleavage  lines  may  be  drawn  by  inserting  the  ocular. 

420.  Becke  Method  for  Determining,  by  Means  of  the  Curvature  of  the 
Isogyres,  the  Value  of  the  Axial  Angle  in  Sections  which  show  the  Point 
of  Emergence  of  but  a  Single  Optic  Axis  (1905). — Not  only  may  the  value  of 
the  optic  angle  be  determined  when  both  melatopes  appear  in  the  field  of  view, 
but  this  may  be  done  when  but  a  single  one  appears.  Becke1  has  shown  that 
the  curvature  of  the  bar  depends  upon  the  value  of  the  axial  angle.  In  the 

determination,  use  is  made  of  the  camera  lucida 
/  and  the  Becke  revolving  drawing-board.     The 

first  step  is  to  indicate  the  direction  of  the 
principal  section  of  one  of  the  nicols  by  laying, 
upon  the  drawing-stage,  a  visiting  card  with  its 


\ 

\ 


T 

J_LJ — L_  edge  parallel  to  the  horizontal  cross-hair.     This 


is  necessary  since  the  interference  figure  is  to  be 
viewed  by  the  Lasaulx  method,  consequently  the 
/  ocular  with   its   cross-hairs  is  to  be  removed. 

FIG.  689.-(A£ter  Becke.)  The  ocular  is  removed  and  the  microscope-stage 
is  rotated  until  the  isogyre  lies  parallel  to  the 

edge  of  the  visiting  card,  as  seen  by  the  aid  of  the  camera  lucida.  In  this 
position  the  trace  of  the  optic  axes  lies  parallel  to  the  principal  section  of  one 
of  the  nicols,  consequently  both  melatopes  must  lie  somewhere  along  the  dark 
bar.  The  card  is  now  removed  from  the  drawing-stage,  and  the  latter  is 
turned  until  its  vernier  shows  the  same  reading  (a)  as  does  the  microscope- 
stage.  In  this  position  the  horizontal  isogyre  is  sketched  on  the  stage. 

Both  microscope-  and  drawing-stage  are  now  rotated,  clock-wise,  through 
45°  (reading  b),  and  the  isogyre  is  sketched  in  its  new  position.  The  point  in 
the  drawing  where  the  two  lines  cross  is  the  point  of  emergence  of  one  of  the 
optic  axes  (A,  Fig.  689). 

The  next  step  is  a  rotation  through  180°  of  the  microscope-  or  drawing- 
stage,  one  or  the  other,  and  the  process  of  drawing  the  isogyre  in  two  positions 
is  repeated.  In  this  manner  a  second  point  A '  is  obtained.  Half  the  distance 
A  A'  determines  C,  the  center  of  the  field. 

With  C  as  a  center,  a  circle  is  drawn  with  a  radius  r  -  — ~ — ,  in  which  /? 

is  the  mean  refractive  index  of  the  mineral,  K  the  Mallard  constant  for  the 
combination  of  lenses  used,  and  p  a  value  so  chosen  that  the  circle  will  not  lie 
too  near  the  edge  of  the  field  of  view,  for  here  the  bars  are  too  broad  and  the 
lenses  themselves  partially  polarize  the  light.2  With  Fuess  objective  No.  7, 

1  F.  Becke:    Messung  des  Winkels  der  optischen  Achsen  aus  der  Hyperbelkrummung. 
T.M.P.M.,  XXIV  (1905),  35-44- 

2  Art.  357. 


ART.  420] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


481 


p  =  20°  is  a  good  value  to  use.     The  circle  just  drawn  will  cut  the  hyperbolae 
in  the  points  H  and  H' '. 

It  is  now  necessary  to  draw  the  points  A  and  H  in  stereographic  projection, 
which  is  most  readily  done  by  the  use  of  transparent  paper  and  a  Wulff  net. 
The  point  of  emergence  of  the  optic  axis  is  located  by  means  of  its  azimuth 
and  central  distance.  The  former  is  determined  by  laying  the  projection 
drawing  upon  the  stereographic  net  in  such  a  position  that  the  lines  o-o 
and  90-90 — drawn  through  C  and  perpendicular  and  parallel  to  the  hori- 
zontal isogyre — correspond  with  the  central  meridian  and  the  equator  of  the 
net.  By  placing  a  straight  edge  along  AA'  (Fig.  689)  the  angle  which  CA' 
makes  with  the  vertical  can  be  read  from  the  graduations  around  the  net. 
This  gives  the  azimuth  of  the  desired  point.  The  central  distance  A  A',  meas- 
ured with  a  millimeter  scale,  is  proportional  to  2D  and,  since  D  =  K  sin  E 

=  K(3swV,we  have  V=-^^}  where  V  is  the  central  distance  in  degrees,  K  the 

Mallard  constant  for  the  instrument,  and  0  the  mean  refractive  index  of  the 
plate.     As  a  control  one  may  measure  the  distance  between  the  horizontal 

isogyre  and  the  center  ( =  20),  whereby  sin  o>  =  -^r,  in  which  co  is  the  angular 

distance  between  the  axial  plane  and 
the  center. 

The  point  H  is  similarly  transferred 
for  azimuth.  Its  central  distance  (p) 
is  known  by  the  construction. 

Having  located  the  points  A  and  H 
on  the  tracing  paper,  the  drawing  is 
rotated  about  the  center  over  the  Wulff 
net  until  the  horizontal  diameter  is 
parallel  to  the  horizontal  isogyre 
through  A  i  (Fig.  690).  The  great  cir- 
cle is  then  drawn  through  A  i  and  the 
ends  of  the  horizontal  diameter.  This 
is  the  trace  of  the  plane  of  the  optic 

axes,  since  it  contains  one  of  the  points  of  emergence  of  the  optic  axes  and 
lies  parallel  to  the  principal  section  of  one  of  the  nicols. 

It  is  necessary,  next,  to  determine  the  vibration  direction  of  the  ray  at  H.1 
According  to  the  Biot-Fresnel2  law,  the  extinction  directions  in  any  section 

1  See  simplified  method  at  the  end  of  this  Article. 

2  J.  B.  Biot:    Memoire  sur  les  lois  generates  de  la  double  refraction  et  de  la  polarisation 
dans  les  corps  regidieremcnt  cristallisees.     Mem.  Acad.  France,  Annee  1818.     Ill  (1820), 
177-384,  especially  228. 

A  Fresnel:  Memoire  sur  la  double  refraction.     Mem.  Acad.  France,  VII  (1827),  45-176. 
Idem:  Ueber  die  doppelte  Strahlenbrechung.     Translation  of  preceding.     Pogg.  Ann., 
XXIII  (1831),  372-434,  494-56c,  especially  542-545- 
31 


FIG.  690. — (After  Becke.) 


482  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  420 

of  a  biaxial  crystal  are  parallel  to  the  traces,  on  that  section,  of  the  planes 
bisecting  the  angles  between  the  two  planes  determined  by  the  optic  axes  and 
the  normal  to  the  section.  Since  the  extinction  directions  are  shown  by  the 
directions  of  vibration  of  the  nicols,  it  is  only  necessary  to  draw,  through  H, 
a  line  parallel  to  the  trace  of  the  polarizer.  This  line  may  readily  be  deter- 
mined since  its  inclination  to  the  horizontal  line  of  the  net  is  equal  to  the 
amount  which  the  stage  has  been  rotated  to  produce  the  isogyre  through 
A\H;  in  the  present  case  45°.  A  line,  therefore,  is  drawn  through  H  inclined 
45°  to  the  horizontal.  The  stereographic  projection  of  the  vibration  plane 
through  H,  however,  will  not  be  this  straight  line,  but  a  great  circle  tangent 
to  it  at  H.  It  may  readily  be  drawn  by  rotating  the  WulfT  net  until  a  great 
circle  is  tangent  at  H.1  Two  points,  E  and  F,  are  now  laid  off,  90°  from  H, 
and  a  great  circle  is  drawn  through  these  points.  Since  this  circle  is  the  polar 
circle  to  H,  it  is  laid  off  by  rotating  the  net  until  H  lies  on  the  equator,  and 

tracing  the  meridian  through  F.  On  this 
meridian  a  distance  FG  =  EF  is  laid  off,  and 
the  great  circle,  drawn  through  GH,  is  the 
trace  of  the  plane  passing  through  the  other 
optic  axis.  But  it  is  already  known  that  the 
plane  BA  iC  is  the  plane  of  the  optic  axes,  con- 
sequently the  unknown  axis,  lying  in  both  the 
BAiC  and  GH  planes,  must  be  at  their  inter- 
section A  2.  This  point,  therefore,  is  the  point 
of  emergence  of  the  other  optic  axis.  The  axial 
angle,  AiA2  =  2V.  may  be  read  directly  from 

FIG.  691.— (After  Becke.)  '  J  J 

the  stereographic  net. 

The  error  of  observation,  according  to  Becke,  is  about  i°  in  the  value  of 
2V,  an  error  of  small  consequence  for  practical  purposes.  Observations 
should  be  repeated  in  a  position  at  180°  from  the  first,  to  eliminate  errors  of 
eccentricity  of  the  instrument,  and  the  mean  values  of  the  four  sets  of  readings 
should  be  transferred  to  the  stereographic  projection.  The  amount  of  rota- 
tion of  the  stage  to  obtain  the  best  position  for  H  depends  upon  the  situation 
of  the  melatope  upon  the  stage.  The  angle  AiHA?.  should  be  neither  too 
acute  nor  too  obtuse,  and  the  angle  HA2B  not  too  small.  It  has  been  found 
that  if  the  melatope  is  too  near  the  center  of  the  field,  the  base  A  iH  will  be 
too  small.  If  the  melatope  falls  too  near  the  margin  of  the  field,  the  length 
of  the  isogyre  seen  is  too  short  to  determine  the  position  at  which  it  is  parallel 
to  the  vibration  plane  of  the  nicol.  Here,  also,  the  polarization  effect  of  the 
edge  of  the  lens  acts  as  a  disturbing  factor.  The  most  satisfactory  position 
is  when  the  horizontal  isogyre  lies  at  a  distance  of  between  one-half  and  one- 

1  The  pole  of  this  great  circle  (HF)  must  lie  on  a  line  through  H,  normal  to  the  tangent, 
and  90°  distant.  The  stereographic  net  is  rotated  until  the  equator  passes  through  P. 
The  required  great  circle  is  the  meridian  of  the  net  now  passing  through  H. 


ART.  421]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  483 

third  of  the  radius  of  the  field  from  the  center,  and  the  acute  bisectrix  lies 
in  one  half  of  the  field  and  one  of  the  optic  axes  in  the  other. 

In  a  later  paper,  Becke1  simplified  the  method  of  finding  the  vibration 
direction  of  the  ray  through  H  by  a  construction  similar  to  that  previously 
proposed  by  Wright.2  The  location  of  the  point  differs  slightly  from  that 
obtained  by  the  latter.3 

The  new  method  is  much  more  quickly  performed  and,  if  repeated  in  the 
1 80°  position,  there  is  less  chance  for  error.  The  great  circle  KL  (Fig.  691), 
polar  to  H,  is  drawn  as  before,  as  is  also  the  vibration  direction  of  thenicol  OX. 
H  is  then  connected  by  a  straight  line  with  AT.  Where  it  cuts  the  great  circle 
KL  is  the  desired  point  F. 

421.  Wright's  Modification  of  the  Becke  Method  for  Determining  the 
Axial  Angle  by  Means  of  the  Curvature  of  the  Isogyres  (1907). — The  principle 
used  by  Becke  for  determining  the  value  of  the  axial  angle  by  means  of  the 
curvature  of  the  isogyres,  was  also  employed  by  Wright,4  but  instead  of 
using  a  revolving  drawing  table,  he  used  a  double  screw  micrometer  ocular 
(Fig.  386).  This  is  an  instrument  in  which,  in  place  of  the  single  movement 
of  the  usual  screw  micrometer  oculars,  there  are  two  movements,  at  right 
angles  to  each  other,  whose  readings  determine  the  position  of  any  point  of 
the  interference  figure,  and  correspond  to  rectilinear  coordinates  in  the  ortho- 
graphic projection  or  small  circle  coordinates  in  the  stereographic.  By  means 
of  the  constant  K  of  the  microscope,  which  must  have  been  determined  pre- 
viously for  each  of  the  movements,  each  reading  is  reduced  to  its  angular 

value  by  the  formula  sin  V  =  ^~,  in  which  /3  is  the  mean  refractive  index  of 

A/3 

the  mineral. 

The  readings  are  made  as  follows:  The  microscope  stage  is  rotated  until 
the  dark  isogyre  is  parallel  to  the  horizontal  cross-hair  of  the  ocular ;  the  hori- 
zontal cross-hair  is  moved  by  means  of  the  vertical  micrometer  screw  V  until 
it  coincides  exactly  with  the  center  of  the  dark  axial  line  (AiC,  Fig.  692), 
and  the  nicols  (not  the  stage)  are  rotated  about  a  suitable  known  angle,  for 
example,  30°  to  45°.  The  optic  axis  A  i  now  corresponds  with  the  intersection 
of  the  isogyre  with  the  horizontal  cross-hair  (HA\  with  A\C).  The  vertical 
cross-hair  is  next  moved  by  means  of  the  horizontal  micrometer  screw  until 
it  coincides  with  this  intersection.  The  two  readings  are  recorded,  after 

1  F.  Becke:    Zuf  Messung  des  Achsenunnkels  aus  der  Hyperbelkriimmung.     T.M.P.M., 
XXVIII  (1909),  290-293. 

2  Fred.  Eugene  Wright:     The  measurement  of  the  optic  axial  angles  of  minerals  in  the 
thin   section.     Amer.  Jour.  Sci.,  XXIV  (1907),  331-341. 

Idem:  Das   Doppel-Schrauben-Mikrometer-OkuJar.     T.M.P.M.,   XXVII    (1908),    293- 

314. 

3  See  Art.  421  injra  and  Fig.  691. 

4  Op.  cit. 


484 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  421 


sm  r 


FIG.  692. 


which  the  stage  is  rotated  through  an  angle  of  180°,  and  similar  readings  are 
taken  for  A  i  in  its  new  position  A  \  to  determine  the  exact  center  of  the  field. 
This  point  (O)  lies  half-way  between  CC'  and  A iA\. 

AI,  being  thus  fixed,  its  position  can  be  plotted  in  stereographic  projection 
by  reducing  the  values  to  the  true  angles  within  the  crystal  by  the  formula 

=  n.     Another  point  H  of  the  isogyre  is  now  determined  by  a  single  set 

of  two  readings,  thus  giving  coordinates 
from  the  center.  These  are  likewise  re- 
duced to  their  true  angles  and  are  plot- 
ted on  the  projection. 

Having  located  the  points  A  i  and  H, 
the  optic  angle  may  be  determined  as 
in  the  Becke  method  or  by  the  follow- 
ing, which  differs  from  the  former  in  the 
manner  of  determining  the  position  of 
the  vibration  direction  through  H. 
According  to  Wright,  this  vibration 
direction  is  the  great  circle  through 
H  and  C  (Fig.  693) .  The  latter  is  deter- 
mined by  the  intersection  of  the  great 
circle  PK — polar  circle  to  H — and  the 
trace  of  the  principal  plane  FOI  of  the 
lower  nicol. 

Having  determined  the  point  C,  the 
distance  A'^C  is  laid  off  equal  to  A'\C. 
The  intersection  of  the  great  circle 
through  H  and  A'2  with  the  great  circle 
DE  gives  the  location  of  the  second 
melatope  A^. 

The  difference  in  position  of  the 
point  C,  as  located  by  Wright  and  by 
Becke,  may  be  seen  from  Fig.  691  in  which  C  is  the  location  by  Wright's 
method  and  F  by  Becke's.  Wright1  claims  that  the  vibration  for  any  dark 
spot  of  the  isogyre  must  be  parallel  to  the  extinguishing  plane  of  the  upper 
nicol,  regardless  of  the  fact  that  two  corresponding  points  darkened  by  vibra- 
tions at  right  angles  to  each  other  do  not  lie  90°  apart,  a  condition  deemed 
essential  by  Becke.2  In  practice,  the  points  located  by  the  two  methods  fall 
so  close  together  that  the  accuracy  of  the  two  is  about  equal,  both  being  ap- 
proximations to  the  true  position.  The  disturbing  element  of  the  rotation 


y 

FIG.  693. 
FIGS.  692  and  693. — (After  Wright.) 


1  Fred.  Eugene  Wright:    The  methods  of  pelrographic-microscopic  research. 
Publication  No.  158,  Washington,  1911,  160. 

2  F.  Becke:  Op.  cit.  T.M.P.M.,  XXVIII  (1909),  290-293. 


Carnegie 


ART.  424]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  485 

of  the  polarized  light  by  the  lenses  and  slide,  and  the  indistinctness  of  the 
isogyres,  produce  greater  errors  than  those  caused  by  the  location  of  the 
points. 

Instead  of  the  double-screw  micrometer  ocular,  an  ocular  with  a  coordinate 
micrometer  scale  (Fig.  384)  may  be  used.  If  the  graduations  are  made  to  o.i 
mm.,  it  will  give  results  nearly  as  accurate  as  the  former  and  is  much  less 
expensive.  The  graduations  should  cover  the  entire  field  given  by  the  Ber- 
trand  lens,  and  should  be  calibrated  in  the  same  manner  as  the  screw  microm- 
eter ocular..  By  the  use  of  cross-section  paper  the  isogyres  may  be  plotted 
directly  for  any  angle  of  rotation. 

A  simple  method  of  calibration,  independent  of  Mallard's  formula,  is  that 
which  makes  use  of  a  Zeiss  apertometer  (Fig.  2 14),  by  means  of  which  it  is  only 
necessary  to  determine  the  number  of  divisions  of  the  scale  covered  by  the 
different  angles. 

422.  Modifications  of  Becke's  Method.     Wright  (1907). — Various  modi- 
fications of  the  Becke  method  for  determining  the  optic  angle  have  been 
proposed. 

In  place  of  a  revolving  stage  fixed  to  a  board  beneath  the  microscope, 
Wright1  attached  a  small  revolving  stage  directly  to  the  stand. 

423.  Stark  (1908). — Stark,2  following  a  suggestion  from  Becke,  did  not 
use  the  revolving  stage,  but  rotated  the  polarizer  and  a  cap  nicol  above  the 
camera  lucida  through  equal  angles.    Later  he  used  a  microscope  with  simul- 
taneously rotating  nicols,  thus  expediting  the  determinations  and  eliminating 
the  error  of  centering.     He  made  his  drawings  by  camera  lucida,  as  in  the 
Becke  method,  and  likewise  transferred  them  to  stereographic  projection.     He 
claims  the  method  requires  but  one- third  the  time  necessary  to  make  readings 
with  the  double  screw  micrometer  ocular. 

424.  Tertsch  (1910). — Tertsch3  eliminated  all  errors  produced  by  paral- 
lax and  by  lack  of  parallelism  between  microscope-  and  drawing-stage,  and 
between  microscope  axis  and  reflected  ray,  by  inserting  a  long  focus  lens  in  the 
tube.     This  projects  a  real  image,  enlarged  and  inverted,  to  the  end  of  the 
tube,  where  it  is  received  on  tracing  paper  placed  over  an  ocular  made  similar 
to  a  cap  nicol,  with  a  circle  divided  to  5°  and  a  vernier  reading  to  degrees. 
The  paper  is  placed  over  the  top  and  is  held  in  place  around  the  edges  by  a 
slip-over  ring.     It  lies  exactly  in  the  plane  of  projection  of  the  interference 
figure  and  therefore  may  be  traced  with  a  pencil.     If  outside  light  is  shut  off 
from  the  top  of  the  microscope  by  means  of  a  hood  or  cloth,  as  is  done  in  a 

1  Fred.  Eugene  Wright:    The  measurement  of  the  optic  axial  angle  of  minerals  in  the  thin 
section.    Amer.  Jour.  Sci.,  XXIV  (1907)  331,  and  Fig.  7,  page  33?. 

2  Michael    Stark:    Geologisch-petrographische    Aufnahme    der    Euganeen.     T.M.P.M., 
XXVII  (1908),  412-413. 

3  Hermann  Tertsch:    Ein  ncues  Zeichenokular.     T.M.P.M.,  XXIX  (1910),  171-172. 


486  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  424 

photographic  camera,  the  figure  may  be  seen  much  more  clearly.  This  appa- 
ratus has  the  advantage  over  the  revolving  table  in  its  cheapness,  in  the  use  of 
white  paper  and  fine  pencil,  in  simplicity,  and  in  rapidity  of  use.  Its  chief 
disadvantage  is  the  reduction  of  light  in  transmission  through  the  paper, 
especially  noticeable  with  small  minerals  whose  interference  figures  require 
small  diaphragms.  If  a  piece  of  ground  glass,  slightly  oiled,  were  used  instead 
of  paper,  the  loss  of  light  would  not  be  so  great. 

In  an  earlier  paper,  Tertsch1  described  a  method  for  determining,  with 
the  Becke  drawing  table,  the  axial  angle  in  sections  cut  at  right  angles  to  a 
bisectrix.  The  method  is  hardly  more  accurate  than  that  given  by  Michel- 
Levy,2  and  is  much  more  complicated. 

1  Hermann  Tertsch:     Versuch  einer  Achsenwinkelmessung  in  einem  Mittellinienschnitt 
T.M.P.M.,  XXVII  (1908),  589-594. 

Review  by  St.  Kreutz:  Zeitschr.  f.  Kryst.,  XLIX  (1910-11),  291-2. 

2  Art.  416. 


CHAPTER  XXXV 

MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  BY  MEANS  OF  A 
ROTATION  APPARATUS 

425.  The  Rotation  Apparatus. — Various  forms  of  rotation  apparatus  and 
the  general  method  for  the  orientation  of  mineral  sections  have  been  described 
in  an  earlier  chapter.  It  now  remains  to  show  the  applicability  of  these 
instruments  to  the  measurement  of  extinction  and  axial  angles.1 

As  mentioned  previously,  modern  rotation  apparatus  or  universal  stages 
are  due  almost  entirely  to  the  work  of  Klein  and  of  von  Fedorow.  The  work 
of  the  former  was  confined  more  especially  to  instruments  adapted  to  the  ex- 
amination of  single  crystals,  while  that  of  the  latter  was  to  instruments  for 
the  examination  of  minerals  in  rock  sections.  Ordinarily  it  is  necessary  to  ex- 
amine a  considerable  number  of  differently  orientated  grains  of  the  same  min- 
eral in  order  that  all  of  its  properties  may  be  determined.  While  this  is  usually 
a  simple  enough  procedure,  it  sometimes  happens  that  but  a  single  fragment 
of  a  mineral  occurs,  or  it  may  be  that  one  is  unable  to  determine  whether  or 
not  a  particular  grain  in  the  slide  is  the  same  as  some  other.  In  such  cases,  if 
one  can  tilt  the  section  to  a  different  angle,  the  effect  is  that  of  having  a 
differently  orientated  section.  Further,  in  certain  cases,  as  for  example  in 
the  determination  of  the  feldspars,  one  may  desire  to  obtain  the  maximum 
extinction  angle  in  a  crystal.  With  a  rotation  apparatus  it  is  possible,  by  a 
slight  inclination  of  the  section,  to  determine  whether  or  not  the  angle  is  at 
its  maximum.  Ordinarily,  when  sufficient  grains  of  the  same  mineral  are 
present  in  the  slide,  the  Fedorow  methods  will  not  be  used,  at  least  in  full, 
but  under  certain  conditions,  they  offer  the  only  possible  solution  to  the  de- 
termination. Some  of  the  methods  are  simple  and  quickly  applied,  while 
others  are  complicated  and  may  require  a  great  deal  of  time,  perhaps  hours, 
for  a  single  mineral. 

Observations  with  the  universal  stage  are  usually  made  by  parallel  and 
not  by  convergent  light,  which  makes  it  possible  to  use  lower  power  objectives 
and  to  cover  a  larger  field.  The  rotation  instrument  is  fixed  on  the  stage  of 
the  microscope  in  such  a  position  that  the  outer  horizontal  axis  (/,  Fig.  695) 
is  parallel  to  the  principal  section  of  one  of  the  nicols,  ordinarily  parallel  to 
the  left-to-right  cross-hair.  The  inner  disk  being  a  glass  plate,  accurately 
divided  into  quadrants,  it  is  easy  to  fix  the  universal  stage  in  proper  position; 

1  Besides  the  papers  mentioned  below  see  also  L.  Duparc  et  R.  Sabot:  Les  methodes 
de  Fedorow.  Arch.  d.  Sci.  Phy's.  et  Nat.  Geneve,  XXXIV  (1912),  (Juillet),  pp.  12. 

487 


488  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  425 

it  is  only  necessary  to  make  these  intersecting  lines  coincide  with  the  cross- 
hairs of  the  microscope.  If  the  microscope  has  a  mechanical  stage,  the 
determination  as  to  whether  the  central  plate  is  truly  at  right  angles  to  the 
axis  of  the  microscope  is  made  by  moving  the  universal  stage  across  the  field 
in  a  direction  at  right  angles  to  the  axis,  and  noting  whether  the  scratch  in  the 
glass  remains  sharply  in  focus. 

Glass  hemispheres  are  sometimes  used  to  increase  the  angle  of  vision.1 
In  size  they  are  as  much  less  than  perfect  hemispheres  as  the  thickness  of  the 
glass  stage  and  the  object-glass,  so  that  when  attached  above  and  below  the 
section  by  a  thin  film  of  glycerine  or  cedar  oil,  their  outer  contours  form  a 
perfect  sphere.  Thus  if  the  glass  plate  of  the  central  stage  is  just  i  mm.  in 
thickness  and  the  object-glass  of  the  preparation  is  the  same,  the  latter 
is  turned  cover-glass  downward,  and  the  two  hemispheres  are  each  i  mm.  less 
in  thickness  than  a  perfect  hemisphere.  For  the  small  apparatus  (Fig.  405), 
in  which  no  glass  stage  is  used  and  the  upper  converging  lens  is  placed 
directly  upon  the  cover-glass  of  the  preparation,  the  upper  hemisphere  is 
almost  perfect  while  the  lower  one  is  cut  down. 

To  set  the  segments  properly,  the  lower  one  is  first  put  into  position  by 
attaching  it  with  a  very  thin  film  of  glycerine  or  cedar  oil,2  the  latter  being 
rather  more  sticky.  Upon  looking  through  the  microscope  with  a  low-power 
objective,  a  bright,  central  circle  of  light  is  seen  surrounded  by  a  dark  circle, 
the  latter,  with  higher  powers,  lying  beyond  the  field  of  view.  The  lower 
hemisphere  should  now  be  moved  laterally  until  the  bright  ring  lies  con- 
centric with  the  field  of  the  microscope.  The  rock  section  is  next  placed 
upon  the  stage,  and  the  mineral  to  be  tested  is  centered.  Careful  note  is 
taken  of  the  exact  part  of  the  mineral,  perhaps  marked  by  a  small  inclusion, 
coinciding  with  the  cross-hairs.  The  upper  glass  segment  is  now  attached 
with  glycerine  or  cedar  oil.  In  general,  a  displacement  of  the  mineral  will  be 
noticed.  The  segment  is  moved  laterally  until  the  original  grain  is  centered, 
that  is,  until  there  is  no  displacement.  If  the  stage  is  now  rotated  in  altitude 
the  mineral  grain  should  remain  exactly  on  the  cross-hairs,  for  it  lies  at  the 
exact  center  of  the  sphere,  through  which,  also,  all  the  rotation  axes  of  the 
microscope  and  the  universal  stage  pass. 

For  the  rapid  centering  of  the  glass  hemisphere,  as  is  desirable  in  examining 
grains  to  determine  their  uniaxial  or  biaxial  character,  they  may  be  set  in 

1  E.  von  Fedorow:  Optische  Mittheilungen.     Noch  ein  Schritt  in  der  Anwendung  der 
Unhersalmethode  zu  optischen  Studien.     Zeitschr.  f.  Kryst.,  XXV  (1895-6),  353. 

Idem:  Uniiersalmethode  und  Feldspathstudien.  /.  Methodische  Verfahren.  Zeitschr. 
f.  Kryst,  XXVI  (1896),  229-231. 

2  If  cedar  oil  is  too  thin,  it  may  be  rendered  much  less  fluid  by  spreading  it  out  in  thin 
layers  and  exposing  it  for  a  long  time  to  the  influence  of  air  and  light.     By  this  means  it 
becomes  of  the  consistency  of  castor-oil  without  increasing  in  dispersive  power.     The 
refractive  index  is  also  raised  to  1.518-1.520.     The  index  can  be  reduced  to  1.510  if  desired, 
by  adding  olive  or  castor-oil.     (E.  Abbe.  Botan.  Centralbl.,  X  (1882),  224-225.) 


ART.  42HJ  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  489 

carriers1  which  are  attached  to  screws  at  the  side  of  the  inner  stage.  By 
this  means  they  must  necessarily  be  placed  in  proper  position.  It  is  better 
practice  to  attach  the  lower  lens  by  glycerine  or  cedar  oil,  since  it  need  not 
be  removed  upon  placing  a  different  mineral  under  the  cross-hairs,  and  only 
use  a  carrier  with  the  upper  segment.  For  fine  determinations,  however, 
it  is  better  to  use  the  loose  hemispheres,  the  field  being  clearer  and  larger  on 
account  of  the  liquid  film  used  for  their  attachment. 

The  size  of  the  field  of  view  will  depend  upon  the  refractive  index  of  the 
glass  segments;  the  greater  the  index,  the  greater  the  angular  view.  Fedorow 
used,  originally,  glasses  with  an  index  of  1.7469,  which  is  higher  than  that  of 
most  rock-forming  minerals,  consequently  their  values  for  2E  were  less  than 
for  2V.  Glass  of  such  high  index  is  very  expensive,  and  the  hemispheres 
usually  provided  with  the  instrument  have  indices  of  1.5233.  They  are,  how- 
ever, just  as  good  for  the  great  majority  of  rock-forming  minerals. 

The  object-glasses  for  mounting  the  preparations,  as  used  by  Fedorow,2 
are  2  cm.  in  diameter  and  circular  instead  of  rectangular.  For  this  special 
purpose  they  possess  the  advantage  of  permitting  every  portion  of  the  slide 
to  be  examined,  and  yet  do  not  interfere  with  the  free  rotation  of  the  inner 
stage.  To  preserve  such  sections,  they  are  kept  in  boxes  into  which  are 
placed  cardboard  strips,  i  mm.  in  thickness 
(same  thickness  as  the  slides),  cut  as  shown 
in  Fig.  694.  Between  these  strips  are  placed  FlG-  694-— Septum  in  the  von  Fedo- 

.,,  i  r  i  ,1  ,  •  row  slide  boxes. 

thin   rectangular   sheets  to  keep  the  sections 

apart,  each  fifth  one  being  thicker  than  the  others  to  aid  in  counting  and  to 
permit  the  writing  of  a  number  on  its  edge.  With  the  Fuess  theodolite  mi- 
croscope,3 sections  28X48  mm.  may  be  used. 

426.  Locating  one  Optic  Axis. — To  determine  the  position  of  the  point 
of  emergence  of  an  optic  axis4  in  its  relation  to  the  normal  to  the  section,  the 
universal  stage  is  set  upon  the  stage  of  the  microscope,  and  is  placed  in  hori- 
zontal position  with  the  axis  /  (Fig.  695)  from  left  to  right  and  the  axis  H 
exactly  at  right  angles  to  it.  The  section,  now  in  a  random  position  with 
respect  to  the  orientation  of  its  vibration  axes,  is  rotated  about  the  axis  H 
until,  between  crossed  nicols,  darkness  ensues.  Should  there  be  no  position 
in  which  it  is  possible  to  produce  darkness  by  rotation  about  the  H  axis,  the 

1  C.  Leiss:  Vervollstandigte  neue  Form  des  E.  v.  Fedorow' schen  Uniiersaltisches.  Xeues 
Jahrb.,  1897  (II),  93-94. 

3  E.  von  Fedorow:  Universal  methode  und  Feldspathstudien.  I.  Methodische  Verfahren. 
Zeitschr.  f.  Kryst.,  XXVI  (1896),  227. 

Idem:  Universalmethode  und  Feldspathstudien.  III.  Die  Feldspathe  des  Bogoslowsk'schen 
Bergreviers.  Zeitschr.  f.  Kryst.,  XXIX  (1897-8).  617-618. 

3  Art.  184. 

4  E.  von  Fedorow:  Universal-  (Theodolith-)  Methode  in  der  Miner alogie  und  Petrographie. 
II.  Theil.     Krystalloptische  Untersuchungen.     Zeitschr.  f.  Kryst.,  XXII  (1893-4),  232. 

Idem:  Op.  cit.  Zeitschr.  f.  Kryst.,  XXVI  (1896),  242-243. 


490 


MANUAL  QF  PETROGRAPHIC  METHODS 


[ART.  426 


FIG.  695. —Von  Fedorow  universal  stage,  large  mode 
(Fuess.)     See  Fig.  407  for  an  improved  form. 


inner  disk  S,  carrying  the  mineral  section,  is  rotated  through  a  not  too 
small  angle,  and  the  operation  is  repeated.  The  section  is  now  placed  as 
nearly  as  possible  in  the  position  of  darkness  by  rotation  about  H.  It  is 
evident  that  the  trace  of  one  of  the  principal  sections  of  the  optical  ellipsoid 
now  lies  parallel  to  the  vibration  plane  of  one  of  the  nicols.  In  general,  how- 
ever, this  plane  of  the  ellipsoid  will  not  be  parallel  to  the  axis  of  the  micro- 
scope, consequently  upon  ro- 
tating the  mineral  section 
about  the  axis  /,  the  trace  will 
become  inclined  to  the  cross- 
hairs.  With  the  mineral  set 
at  some  angle  about  /,  the 
stage  is  tilted  about  H,  and 
notice  is  taken  whether  dark- 
ness appears  with  greater  or 
less  inclination.  One  will  soon 
find  that  for  a  rotation  about 
/  in  one  direction,  darkness 
will  appear  at  a  greater  angle,  and  for  a  rotation  in  the  opposite  direction 
at  a  lesser  angle.  There  is  thus  determined  the  direction  of  rotation  of  the 
preparation  necessary  to  bring  one  of  the  principal  sections  of  its  optical  ellip- 
soid parallel  to  the  principal  section  of  one  of  the  nicols.  When  in  this  posi- 
tion, the  field  will  remain  dark  during  the  rotation  about  /.  If  it  does  not 
quite  do  so,  a  very  slight  rotation  about  H  will  correct  the  error. 

The  stage  of  the  microscope,  or  the 
disk  TI,  may  now  be  rotated,  with  suc- 
cessive settings  about  /,  until  the  point 
is  reached  where  the  stage  remains  com- 
pletely dark  during  this  revolution  also ; 
the  position  of  darkness  being  accu- 
rately determined  by  the  fact  that  a  unit 
retardation  plate  remains  of  uniform 
color  during  the  rotation.  Strictly 
speaking,  such  a  point  is  never  reached 
in  biaxial  minerals,1  owing  to  dispersion, 
but  the  error  is  ordinarily  so  slight  that  it  may  be  neglected.  If  it  is  great, 
monochromatic  light  may  be  used,  or  the  section  may  be  oriented  simply 
by  the  position  of  maximum  darkness. 

In  the  position  of  darkness  one  of  the  optic  axes  (be,  Fig.  696)  has  its 
apparent  direction  (b'c)  parallel  to  the  axis  of  the  microscope.  The  position 
of  the  point  of  its  emergence  is  located  on  the  section  by  two  readings  of 

1  E.  Kalkowsky:  Ueber  die  Polarisationsverhaltnisse  wn  senkrecht  gegen  eine  optische 
Axe  geschnittenen  zweiaxigen  kiystallplatten.  Zeitschr.  f.  Kryst.,  IX  (1884),  486-497. 


FIG.  696. 


FIG.  697. 


ART.  426] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


491 


the  stage.  The  true  angle  of  inclination  of  the  optic  axis  with  respect  to 
the  normal  (nb)  to  the  section  may  be  calculated  from  the  angles  nbc,  nb'c, 
and  the  mean  refractive  index  of  the  mineral.  This  calculation  may  be 
performed  by  means  of  the  formula  n  sin  V  =  sin  E,  where  n  is  the  mean 
refractive  index  of  the  mineral,  V  (nbc)  the  true,  and  E  (nb'c)  the  apparent 
angle  which  the  axis  makes  in  air  with  the  normal.  Several  graphical 


.10,1.20  etc. 

1.05,1.15    " 


FIG.  698. — Graphical  solution  of  n  sin  V  =  sin   E.     (After  von  Pedorow.) 

solutions  of  this  equation  have  been  given,1  the  one  shown  in  Fig.  698  being 
applicable  to  the  determination  of  the  true  angle  whether  the  apparent 
angle  was  measured  in  air,  oil,  glass  (Fig.  697),  or  any  other  medium. 

1  E.  von  Fedorow:  Cit.  supra,  Zeitschr.  f.  Kryst.,  XX  (1893-4),  247-8.  Gives  a  dia- 
gram for  converting  2E  in  air  to  2  V. 

Idem:  Cit  supra,  Zeitschr.  f.  Kryst.,  XXV  (1895-6),  354-5.  Gives  a  diagram  for 
converting  2H  in  glass  to  2V. 

Idem:  Cit.  supra,  Zeitschr.  f.  Kryst.,  XXVI  (1896),  246-247,  and  plate  IV,  Fig.  3. 
Gives  the  diagram  referred  to  above.  The  same  diagram  is  given  by  Fred.  Eugene 
Wright:  The  methods  of  petro graphic-microscopic  research,  Washington,  1911,  plate  7. 


492 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  426 


The  method  of  use  is  as  follows:  Suppcse  a  mineral  with  a  mean  refractive 
index  of  1.5  gave  an  apparent  angle  in  air  of  48°  10'.  Find  this  value  on  the 
circumference  of  the  circle  (index  of  air  =  i.o),  follow  the  radius  to  its  inter- 
section with  the  circle  of  1.5  refractive  index,  and  then  the  horizontal  line  to 
the  right  (Fig.  699)  to  the  degree  marks  at  the  circumference,  in  this  case 

3°°. 

If  the  second  medium  is  glass,  for  example,  with  a  refractive  index  greater 
than  the  mineral,  the  formula  used  is 

sin  H  =  —  sin  V, 

m 

where  n  is  the  refractive  index  of  the  mineral  and  m  that  of  the  second  medium. 
If,  for  example,  the  glass  has  a  refractive  index  of  1.75,  the  value  of  H, 
corresponding  to  F  =  3o°,  in  the  same  mineral  (72  =  1.5)  as  above,  is  25°  25'. 
Using  the  diagram  (Fig.  698),  the  observed  angle  being  25°  25',  follow  the 
radius  from  this  value  to  its  intersection  with  1.75,  follow  the  horizontal  line 
to  the  left  to  its  intersection  with  the  1.5  circle,  and  read  the  angle  at  the  end  of 


Air 


FIG.  699.  FIG.  700.  FIG.  701. 

FIGS.  699  to  701. — Index  sketches  showing  methods  of  using  the  preceding  diagram.  Fig.  699. 
Denser  medium  to  air.  Fig.  700.  Rarer  medium  to  denser.  The  true  angle  is  greater  than  the  ob- 
served. Fig.  701.  Denser  to  rarer  medium.  The  true  angle  is  less  than  the  observed. 

the  intersecting  radius  (Fig.  700).  If  the  refractive  index  of  the  second 
medium  is  less  than  that  of  the  mineral,  follow  the  horizontal  line  to  the  right, 
instead  of  to  the  left,  in  the  same  manner  as  that  used  when  this  medium  is 
air  (Fig.  701).  The  general  rule  to  be  followed  in  every  case  is  to  follow  the 
radius  from  the  observed  value  to  the  curve  representing  the  refractive  index 
of  the  mineral.  The  horizontal  line  passing  through  this  point  is  followed 
to  its  intersection  with  the  curve  of  the  second  medium.  The  angle  desired 
is  obtained  by  following  the  radius  through  the  latter  point  to  the  circum- 
ference of  the  circle.  The  diagram  (Fig.  698)  shows,  likewise,  the  critical 
angle  between  any  two  substances,  this  being  the  point  where  a  horizontal 
line  is  tangent  to  the  circle  of  the  refractive  index  of  the  denser  medium. 
Thus  the  critical  angle  between  water  (n  =  1.335)  and  air  is  found  by  the  inter- 
section of  the  horizontal  tangent  to  the  1.335  curve  and  the  curve  of  air  (i.o). 


ART.  426]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  493 

Its  value  is  48°  30'.  For  crown  glass  (n=  1.608)  the  critical  angle  with  air  is 
38°  30'.  As  between  quartz  (^  =  1.54)  and  water  (^  =  1.33)  the  angle  is  60°. 

Another  method  for  determining  the  location  of  the  principal  vibration 
planes  in  a  crystal  and  the  position  of  the  optic  axes,  was  given  by  Klein1 
in  1895,  and  later  by  Evans.2  The  method  was  intended  to  be  used  for  the 
determination  of  the  optic  axial  angle  in  mineral  sections  immersed  in  a  fluid 
having  a  refractive  index  as  nearly  as  possible  equal  to  that  of  (3  of  the  mineral. 
By  parallel  light  and  crossed  nicols,  the  preparation  is  tilted  until,  as  in  the 
case  previously  described,  the  section  remains  dark  during  a  rotation  about 
the  other  axis.  In  this  position  the  axis  of  rotation  is  the  optic  normal  or 
one  of  the  bisectrices  of  the  optic  axial  angle.  In  the  former  case  the  section 
will  remain  dark  except  at  the  points  where  the  optic  axes  emerge.  Here, 
on  account  of  internal  conical  refraction,  there  will  be  a  very  slight  increase  in 
light,  but  the  intensity  will  remain  the  same  during  a  complete  rotation 
about  the  vertical  axis  M  (axis  of  the  microscope).  The  determination, 
however,  cannot  be  made  very  accurately,  and  it  is  therefore  better  to  rotate 
the  nicols  to  the  45°  position.  The  mineral  will  now  appear  uniformly  light. 
Insert  the  gypsum  plate  and  determine  whether  the  vibrations  along  the  axis 
of  rotation  are  faster  or  slower  than  in  the  direction  at  right  angles  to  it. 
If  the  axis  of  rotation  is  one  of  the  bisectrices,  it  will  be  the  direction  of  great- 
est or  least  ease  of  vibration,  consequently,  if  the  crystal  be  rotated  about 
the  horizontal  axis  of  the  stage,  there  will  be  no  change  in  the  sign  (-f-  or  — ) 
of  the  mineral,  although  the  birefringence  will  vary  from  y—a  to  7—  /3,  or 
7  —  a  to  J3  —  a.  If,  however,  the  axis  of  rotation  is  the  optic  normal  (b),  the 
optic  sign  will  change  four  times  during  the  rotation,  depending  upon  whether 
the  section  which  happens  to  be  horizontal  gives  a  birefringence  of  7— /?  or 
fi  —  a.  That  is,  the  position  of  greater  ease,  in  the  particular  section  which 
happens  to  lie  at  right  angles  to  the  axis  of  the  microscope,  will  first  be  along 
the  axis  of  rotation  and  then  at  right  angles  to  it.  The  optic  axes  emerge 
at  the  points  where  the  change  from  positive  to  negative  character  occurs, 
and  here  the  retardation,  except  for  dispersion,  etc.,  is  zero.  Its  exact  position 
is  given  by  the  gypsum  plate  when  the  sensitive  tint  appears,  or,  better,  by  a 
combination  wedge,  such  as  the  Evans  double-quartz3  or  the  Wright  double- 
combination  wedge,4  when  the  black  bars  of  the  two  halves  coincide  in 
position. 

For  thin  sections,  it  is  advantageous  first  to  find,  by  convergent  light,  a 
section  which  gives  the  emergence  of  the  acute  bisectrix  or  to  locate  the 

1  C.  Klein:    Ein    Universaldrehapparat    zur  Untersuchiing  von   Diinnschliffen  in  Fliis- 
sigkciten.     Sitzb.  Akad.  Wiss.  Berlin,  1895  (II),  1151-1159. 

2  John  W.  Evans:    Determination  of  the  optic  axial  angle  of  biaxial  crystals  in  parallel 
polarized  light.     Mineralog.  Mag.,  XIV  (1905),  157-159. 

3  Art.  315- 

4  Art.  317. 


494  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  427 

melatopes  by  means  of  a  Johannsen  auxiliary  lens  and  a  low  power  objective. 
The  glass  hemispheres  suggested  by  Fedorow  cannot  be  used  in  this  method 
since  the  light  does  not  remain  strictly  parallel  and  this  makes  it  difficult  to 
locate  the  desired  point.  Immersion  is  not  necessary  with  thin  sections. 

427.  Determination  of  the  Position  of  an  Optic  Axis  by  Means  of  the 
Optical  Curves. — The  point  of  emergence  of  an  optic  axis  may  be  determined 
by  plotting,  in  stereographic  projection,  the  curves  of  zero  extinction  in 
certain  zones,1  and  finding  the  point  of  their  intersection.  This  method, 
called  by  von  Fedorow2  the  method  by  optical  curves,  is  best  used  with  a 
microscope  having  simultaneously  rotating  nicols,  although  an  ordinary 
polarizing  microscope  may  be  used.  It  is  based  upon  the  Biot-Fresnel3  law 
which  states  that  the  trace,  on  the  plane  of  the  section,  of  the  plane  bisecting 
the  angle  between  the  two  planes,  each  determined  by  an  optic  axis  and  the 
normal  to  the  section,  is  the  extinction  angle  in  that  section.  To  obtain  the 
curves,  the  nicols  are  first  crossed  and  set  at  some  fixed  angle  in  relation  to 
the  cross-hairs.  The  thin  section  is  placed  in  a  horizontal  position  and  ro- 
tated to  extinction  about  the  M  axis  (axis  of  the  microscope).  This  angle  is 
read,  reduced  to  the  true  angle  by  means  of  diagram  Fig.  698,  and  plotted 
in  stereographic  projection.  The  inner  stage  T\  is  now  turned,  by  successive 
5°  or  10°  rotations,  and  at  the  same  time  there  is  determined,  by  rotation, 
about  the  axis  /,  which  remains  parallel  to  its  original  position,  the  angle 
through  which  it  is  necessary  to  tilt  the  section  to  obtain  darkness.  The 
procedure  may  be  reversed,  and  the  stage  tilted  about  /  by  successive  5°  or 
10°  rotations  and  the  angle  on  T\  determined.  These  values  are  all  plotted 
in  stereographic  projection,  after  being  reduced  to  their  true  values;  the  mean 
value  of  the  refractive  index  (/3)  being  used  instead  of  the  varying  indices 
in  each  different  position  with  no  appreciable  error  in  minerals  having 
weak  or  medium  birefringence.  The  curve  thus  obtained  must  pass  through 
the  point  of  emergence  of  the  optic  axis  of  the  crystal,  and  necessarily  also 
through  the  center  of  the  projection.  It  is  called  the  curve  of  extinction  and 
has  a  fixed  position  for  a  definite  position  of  the  nicols.  If  the  relative  posi- 
tion between  the  latter  and  the  axis  /  is  changed  by  rotating  the  nicols  or 
the  stage  of  the  microscope,  and  the  same  method  of  procedure  followed, 
a  different  curve,  also  passing  through  the  point  of  emergence  of  the  optic 
axis,  is  obtained.  The  intersection  of  two  such  curves  serves  to  locate  the 
desired  point.  As  a  check,  it  is  better  to  determine  three  or  more  curves,  for 
example  with  the  nicols  set  at  o°,  22  1/2°,  and  45°.  Owing  to  slight  inac- 
curacies in  determination,  the  different  curves  ordinarily  do  not  intersect 
in  a  point,  but  form  a  polygon,  the  center  of  which  is  taken  as  the  true  point 

iCf.Art.3Si- 

2  E.  von  Fedorow:    CiL  supra,  Zeitschr.  f.  Kryst.,  XXVI  (1896),  231-9. 

3  See  Art.  351  supra.      The  reason  for  this  extinction  in  inclined  sections  may  appear 
more  clearly  from  the  demonstration  in  the  next  section. 


ART.  428] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


495 


of  emergence.  It  is,  of  course,  not  necessary  to  determine  the  complete 
curves  but  only  that  part  in  the  neighborhood  of  the  optic  axis.  The  curves 
may  be  named  from  the  inclination  of  the  principal  sections  of  the  nicols  to 
the  cross-hairs,  o°,  15°,  30°,  45°,  etc.,  extinction  curves.  In  Fig.  702  the  c° 
curve  (nicols  parallel  to  the  cross-hairs)  is  drawn  out  in  full,  the  22  1/2°  and 
45°  curves  in  part. 

This  method  of  optical  curves  can  be  used  for  the  determination  of  the 
optic  axial  angle  only  when 'the  points  of  emergence  of  both  optic  axes  appear 
in  the  field  of  the  microscope,  in  which  case 
they  may  both  be  located  by  intersections. 
The  values  of  the  angles  of  inclination  hav- 
ing already  been  reduced  to  their  true 
values,  the  measured  distance  on  the  pro- 
jection gives  the  value  of  2V. 

428.  Locating  the  Point  of  Emergence 
of  the  Second  Optic  Axis. — Not  commonly 
will  the  points  of  emergence  of  both  optic 
axes  appear  in  the  field  at  the  same  time,  al- 
though one  may  be  brought  in,  in  the  ma- 
jority of  cases,  by  tilting  the  stage,  and  its  FlG-  702.— Determination  of  the  optic 

,         .    .  -11          T  i       axial    angle    by    the    method    of    optical 

position  may  be  determined  by  direct  ob-  curves. 
servation  or  as  the  point  of  intersection  of 
several  optical  curves;  the  other,  however, 
must  be  located  by  different  means. 

The  simplest  method1  for  determining 
the  second  optic  axis  is  as  follows:  The 
section  is  placed  in  a  horizontal  position 
and  is  then  rotated  about  the  vertical  axis 
(M)  of  the  microscope  by  means  of  the  disk 
TI  (Fig.  695),  until  the  known  optic  axis 
lies  in  the  plane  at  right  angles  to  the  axis 
/  (EOA'j  Fig.  703).  When  in  this  position 
let  HOC  represent  the  extinction  angle. 
By  the  Biot-Fresnel  law,  the  unknown  axis  FlG-  703-— Locating  the  point  of 

gence  of  the  second  optic  axis. 

OB  must  lie  in  the  vertical  plane  OZ),   so 

placed  that  the  angle  COD  =  HOC.  If  the  stage  is  now  tilted  about  the 
axis  /,  the  extinction  direction  OC  must  change  for  every  different  inclina- 
tion, since  by  this  rotation  the  angle  HOB  changes  its  value.  When  the  axis 
OB  lies  in  the  vertical  plane  through  the  /  axis  (OJ  plane)  the  extinction  angle 
OC'  must  be  45°  since  HOB'  =  90°  =  2  HOC'. 

We  have  here,  then,  an  indirect  method  by  which  we  can  determine  the 


1  E.  von  Fedoro\v:  Cit.  supra,  Zeitschr.  f.  Kryst.,  XXVI  (1896),  234-5. 


496 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  428 


melatope  B,  for  when  the  mineral  section  has  been  rotated  about  J  to  the 
point  where  the  extinction  angle  is  45°,  it  lies  in  the  OJ  plane.  The  position 
of  45°  extinction  may  be  obtained  readily  by  setting  the  crossed  nicols  at 
45°  and  rotating  the  mineral  about  /  to  extinction. 

It  is  now  necessary  to  locate  B  in  the  stereographic  projection  from  the 
two  known  positions.  When  the  mineral  section  is  horizontal  we  know  that 
the  first  optic  axis  emerges  at  A',  and  the  second  somewhere  along  OD. 
When  the  mineral  section  is  tilted  about  /  through  the  angle  6',  the  second 
axis  lies  in  OJ.  If  this  plane  (J'OJ)  be  now  tilted  through  an  angle  0' 
(corrected  for  refractive  index  by  Fig.  698)  in  the  opposite  direction  from  that 
in  which  the  stage  was  inclined,  the  mineral  section  will  again  lie  in  the 
horizontal  plane,  and  we  will  have  located  two  planes  in  which  the  unknown 
axis  lies.  The  intersection  gives  the  position  of  the  axis.  In  the  projection 
the  line  OD,  at  twice  the  extinction  angle  from  H,  is  known.  Draw  J'EJ,  the 
great  circle  of  the  plane  inclined  at  an  angle  of  0'  with  the  vertical,  through 

the  point  E.     The  intersection  of  OD  and 
EJ  is  the  point  of  emergence  of  B. 

This  method  cannot  be  applied  to 
such  cases  where  the  position  of  the 
second  axis  requires  a  steeply  tilted  stage. 
The  effect  of  the  elliptical  polarization  in 
such  sections  makes  the  exact  position  of 
extinction  a  matter  of  uncertainty,  and  a 
different  method,  unexpectedly  exact,  was 
used  by  von  Fedorow.1 

This  second  method  is  likewise  based 
on  the  Biot-Fresnel  law.  Instead  of  set- 
ting the  crossed  nicols  in  some  definite 
position  with  respect  to  /  and  determin- 
ing the  inclination  required  to  produce  darkness  in  the  section,  the  prepa- 
ration, after  having  its  known  optic  axis  placed  on  the  HH'  line,  is 
inclined  at  various  angles,  and  the  crossed  nicols  are  rotated  to  determine  the 
extinction  angles.  For  accuracy,  the  readings  are  repeated  two  or  three 
times  for  each  position  of  the  nicols,  which  are  then  rotated  through  90° 
four  successive  times,  and  the  readings  repeated  in  each  position.  The  stage 
is  tilted  up  or  down,  depending  upon  which  side  of  the  axis  /  the  point  of 
emergence  of  the  optic  axis  falls.  Best  results  are  obtained  by  tilting  the 
section  rather  steeply  and  using  the  same  values  on  either  side  of  the  axis. 
For  simplicity  in  plotting,  such  angles  for  tilting  the  section  are  chosen  as  have 
their  true  values,  and  not  their  measured,  in  even  degrees. 

Having  obtained,  in  this  manner,  several  readings,  the  results  are  plotted 
in  stereographic  projection.     The  method  of  plotting  is  similar  to  that  pre- 
XE.  von  Fedorow:  Cit.  supra,  Zeitschr.  f.  Kryst.,  XXVI  (1896),  235-6. 


ART.  429]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  497 

viously  described.  For  example,  let  the  observed  extinction  angle  (that  is 
the  angle  at  which  the  nicols  were  set)  be  a.  Draw  through  the  center  the 
line  DO  (Fig.  704)  so  that  DOE  =  2  a.  DO,  therefore,  is  the  trace  of  the  verti- 
cal plane  containing  the  optic  axis  in  its  rotated  position.  Let  A  be  the  true 
(not  observed)  angle  of  inclination  of  the  section  about  the  axis  /.  Evidently 
when  the  mineral  section  is  revolved  back  to  the  horizontal  position,  every 
point  on  OD  must  revolve  A°  in  planes  perpendicular  to  /.  Since  these 
vertical  planes  are  represented  by  the  vertical  small  circles  of  the  net,  it  is 
only  necessary  to  lay  off  A°  (30°  in  the  figure)  in  the  proper  direction  along  the 
vertical  small  circles,  and  connect  these  points  by  a  great  circle. 

By  thus  constructing  several  great  circles  for  different  angles  of  extinc- 
tion, each  containing  the  desired  optic  axis,  it  is.  clear  that  the  desired  position 
is  at  their  intersection.  Ordinarily  they  do  not  intersect  in  quite  the  same 
point,  but  they  fall  so  closely  together  that  there  is  no  difficulty  in  determin- 
ing the  mean.1 

This  method  is  very  accurate  if  the  position  of  the  first  axis  has  been 
correctly  determined.  If  it  has  not  been,  the  variation  in  the  points  of  inter- 
section of  the  second  axis  at  once  furnish  a  measure  of  the  amount  of  inaccu- 
racy and  its  direction,  thus  permitting  a  relocation  of  the  first  point  and  a 
new  trial  for  the  second.  If  the  great  circles  finally  intersect  in  a  point,  the 
two  axes  are  accurate  to  1/2°. 

Of  all  methods  of  locating  the  optic  axes  by  means  of  the  universal  stage, 
that  of  optical  curves  is  the  most  accurate,  but,  as  von  Fedorow2  himself 
says,  "The  method  is  practically  unavailable  on  account  of  the  length  of 
time  required.  Even  the  method  of  the  direct  determination  of  the  sym- 
metry planes,  which  with  sufficient  practice  may  be  performed  in  not  over 

two  hours causes  too  long  an  interruption  in  practical  petrographic 

determinations." 

An  algebraic  computation  of  the  positions  of  the  two  optic  axes  as  deter- 
mined by  the  optical  curves  is  given  by  Wallerant.3 

429.  Locating  the  Symmetry  Planes  and  the  Axes  of  the  Optical  Ellipsoid 
within  the  Crystal. — -The  methods  previously  given  can  be  used  with  a 
universal  apparatus  having  but  two  axes  of  rotation.  The  following  method 
requires  the  use  of  three  axes,  and  is  more  conveniently  performed  with  four 
(Fig.  407).  By  it  the  positions  of  the  symmetry  planes  are  directly  located, 
and  from  these  the  various  optical  properties. 

1  A  slight  modification  of  this  method,  for  use  when  the  intersecting  angle  is  very  acute, 
is  given  by  Fred.  E.  Wright:  Measurement  of  the  optic  axial  angle  of  minerals  in  the  thin 
section.      Amer.  Jour.  Sci.,  XXIV  (1907),  351-353,  and  in  Methods  of  petrogr aphic-micro- 
sco pic  research,  Washington,  1911,  181-183. 

2  E.  von  Fedorow:    Cit.  supra,  Zeitschr.  f.  Kryst.,  XXIX  (1897-8),  606. 

3  Fr.  Wallerant:    Stir  la  methode  de  determinations  des  axes  optiques  de  M.  E.  v.  Fedorow. 
Bull.  Soc.  Min.  France,  XIX  (1896),  356-363. 

J.  Beckenkamp:  Review  of  above  in  Zeitschr.  f.   Kryst.,  XXIX    (1897-8),  431-432. 
32 


498  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  430 

In  determining  the  symmetry  planes  by  this  method, 1  the  universal  stage 
is  first  placed  in  horizontal  position  with  the  /  axis  at  right  angles  to  the  H 
axis,  and  the  position  of  one  of  the  principal  sections  of  the  optical  ellipsoid 
is  determined  by  the  method  given  in  Art.  426.  In  this  position  the  section 
should  remain  uniformly  dark  during  the  rotation  about  the  axis  /.  Care 
should  be  used  in  determining  correctly  this  position  of  maximum  darkness, 
since  any  error  in  the  result  will  be  from  neglect  in  this  respect.  Having  the 
angle  of  inclination  about  the  horizontal  axis  H,  and  the  angle  of  rotation 
about  the  axis  M  on  the  stage  7\,  the  points  g,  m,  and  p,  representing  lines  in 
the  symmetry  planes  at  right  angles  to  the  axis  of  rotation,  are  to  be  plotted 
in  stereographic  projection,  using,  of  course,  the  true  and  not  the  observed 

angles.  Thus  in  Fig.  705,  the  symmetry 
plane  a(3  is  determined  by  the  angle  Mg, 
indicating  the  rotation  about  the  axis  H, 
and  HMHi,  the  rotation  about  M.  In  a 
similar  manner  the  planes  ay  and  187  are 
determined  by  the  angles  Mm  and  HMH», 
and  Mp  and  HMH^.  Having  determined 
the  points  g,  m,  and  />,  the  great  circles 
representing  the  traces  of  the  symmetry 
planes  may  be  drawn  in  the  projection 
through  these  points,  thus  locating  by 
their  intersections,  the  points  a,  0,  and  7, 
which  represent  the  points  of  emergence 

FIG.   705. — Method  of  locating  the  symmetry       r    .-,        f  •     ,          '      ,.    ,  j      i 

planes  and  the  axes  of  the  optical  ellipsoid.      of  the  fastest,  intermediate,  and  slowest 

rays  of  the  crystal.     As  a  check  on  the 

accuracy  of  the  construction,  it  may  be  noted  (i)  that  the  lines  connecting  a 
and  p,  jS  and  m,  and  7  and  g  should  be  straight  and  should  pass  through  the 
center  M  of  the  projection,  and  (2)  that  the  angular  distances  between  a  and  p, 
/?  and  m,  7  and  g,  a  and  0,  /?  and  7,  and  a  and  7  should  each  be  90°.  The  first 
condition  gives  a  check  upon  the  accuracy  of  the  determinations;  the  second 
upon  the  accuracy  of  the  value  assumed  for  13  in  reducing  the  observed  to  the 
true  angular  values.  By  noting  whether  the  distances  between  a  and  p,  /3  and 
m,  and  7  and  g  are  greater  or  less  than  90°,  it  permits  a  correction  to  be  applied 
to  the  assumed  value  of  0,  and  a  redrawing  of  the  projection.  The  method 
thus  serves  for  the  rough  determination  of  the  value  of  the  mean  refractive 
index. 

430.  Determination  of  the  Position  of  the  Second  Optic  Axis  when  the 
First  is  Determinable  by  Optical  Curves. — If  one  optic  axis  (A )  can  be  deter- 
mined by  means  of  optical  curves,  the  other  one  (B)  is  easily  located  after 
having  found  the  symmetry  planes,  since  ay  must  be  the  plane  of  the  optic 

1  E.  von  Fedorow:  Cit.  supra,  Zeitschr.  f.  Kryst,  XXVI  (1896),  240-244. 


ART.  431] 


MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE 


499 


B'B  B" 


axes,  and  either  a  or  7  must  be  the  acute  bisectrix,  depending  upon  the  optical 
character  of  the  mineral.  In  many  cases  where  it  could  not  otherwise  be 
determined,  the  optical  character  can  be  found  by  means  of  the  Johannsen 
auxiliary  lens,1  used  with  a  low-power  objective  and  a  tilted  stage.  Know- 
ing the  optical  character,  A  a  or  Ay  may  be  made  equal  to  B  a  or  By. 

The  time  required  for  the  determination  of  the  optic  axes  by  means  of 
symmetry  planes  is  about  two  hours.2 

431.  Approximate  Determination  of  the  Optic  Axes  when  the  Section 
lies  nearly  Parallel  to  the  Plane  of  the  Optic  Axes. — To  determine  the  posi- 
tion of  the  optic  axes  when  the  plane  of  the  optic  axes  makes  an  angle  of  not 
over  25°  with  the  plane  of  the  section,  the  following  method  may  be  used.3 
Having  located  the  position  of  0,  it  is  brought  into  coincidence,  by  means  of 
the  axis  H,  with  the  axis  of  the  microscope, 
thus  bringing  the  plane  of  the  optic  axes 
into  the  horizontal  plane.  This  horizontal 
plane  will  not  be  disturbed  by  rotating  the 
stage  Ti  about  the  vertical  axis  M .  By  the 
previous  construction  the  symmetry  planes 
ad  and  yyr  (Fig.  706)  have  been  located, 
the  third  being  the  horizontal  plane.  Let 
the  mineral  section  now  be  rotated  about  M 
on  the  TI  stage  until  one  of  the  optic  axes 
(B)  coincides  with  the  axis  HH' ',  a  position 
determined  by  trial.  Let  y  be  the  acute 
bisectrix,  for  example.  The  other  optic  axis 
will  therefore  occupy  the  position  MA  such  that  BMy=yMA,  My  being 
a  direction  of  extinction.  The  section  is  now  rotated  about  the  axis  / 
through  a  definite  angle  0  (corrected  for  refractive  index)  so  that  the 
point  B  falls  at  E.  The  extinction  angle  for  the  new  position  (BMn)  is 
now  read;  it  should  bisect  the  angle  BMA',  the  latter  being  determined 
from  the  stereographic  projection,  since  it  must  lie  at  the  intersection 
of  the  vertical  small  circle  through  A  and  the  great  circle  through  J'EJ. 
If  the  two  values  do  not  agree,  it  indicates  that  the  first  optic  axis  did  not 
coincide  exactly  with  OB.  If  the  angle  is  larger  than  it  should  be,  it  shows 
that  the  optic  axis  lies  on  the  side  OB ';  if  too  small,  on  the,  side  OB".  The 
section,  therefore,  should  be  rotated  through  small  angles  in  the  proper 
direction,  and  new  sets  of  determinations  made  until  the  observed  and  con- 
structed values  agree.  In  this  position  the  extinction  angle  should  be  deter- 
mined carefully;  twice  its  value  is  the  value  of  2V. 

1  Albert  Johannsen:    An  accessory  lens  for  observing  interference  figures  of  small  mineral 
grains.     Jour.  Geol.,  XXI  (1913),  96-98. 

2  E.  von  Fedorow:    Cit.  supra,  Zeitschr.  f.  Kryst.,  XXIX  (1897-8),  606. 
SE.  von  Fedorow:  Cit.  supra,  Zeitschr.  f.  Kyrst.,  XXVI  (1896),  245-6. 


500 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  432 


This  case  is  the  most  difficult  of  the  methods  for  determining  the  optical 
properties,  and  it  is  therefore  advisable  to  choose  a  different  fragment,  if 
this  is  possible. 

432.  Simplified  Methods. — As  mentioned  above,  the  most  accurate  of 
the  methods  for  determining  optic  axial  angles  by  means  of  the  universal  stage, 
is  by  optical  curves,  but  it  is  practically  unavailable  on  account  of  the  time 
required  for  its  execution.  Even  the  method  by  the  direct  determination 
of  symmetry  planes,  which  requires,  with  sufficient  practice,  not  over  two 
hours,  is  too  slow  for  practical  work.  For  this  reason  von  Fedorow  simplified 
further,  as  much  as  possible,  the  methods  of  determination,  and  used  only 
the  most  accurate  of  the  rapid  methods.  He  gives  the  following:1 

433»  (a)  Both  Optic  Axes  appear  in  the  Field  of  the  Microscope  at  the 
most  Satisfactory  Angle,  namely,  Inclined  between  15°  and  55°  (Corrected 
Values)  with  the  Normal  to  the  Section. — By  means  of  the  rotation  axes 
/  and  My  bring,  as  nearly  as  possible,  the  more  inclined  optic  axis  to  the 
vertical  position,  and  find,  from  the  stereographic  projection  by  means  of 
optical  curves,  its  exact  position.  Determine  whether  the  bisectrix  of  the 
axial  angle  is  parallel  to  a  or  7  by  means  of  the  mica  wedge.  Greater  accuracy 
may  be  obtained  if  the  other  symmetry  planes  likewise  are  brought  into 
position  at  right  angles  to  the  axis  /,  and  corrections  applied  to  the  projection. 

434.  (b)  One  Optic  Axis  makes  an  Angle  of  less  than  20°  with  the 
Normal  to  the  Section. — 'The  extinction  angle,  in  this  case,  will  be  very 

indistinct.  Place  the  universal  stage  with 
the  central  disk  horizontal,  the  M  axis  coin- 
ciding with  the  axis  of  the  microscope,  and 
the  H  axis  at  right  angles  to  /  and  M . 
Turn  the  inner  glass  circle,  and  at  the  same 
time  incline  the  section  on  the  H  axis,  until 
the  optic  axis  coincides  as  nearly  as  possible 
with  the  axis  of  the  microscope.  Now  tilt 
to  a  considerable  angle  on  the  /  axis,  and  at 
the  same  time  rotate  on  the  M  axis  to  dark- 
ness. Since  the  plane  of  the  optic  axes  in 
this  position  is  at  right  angles  to  /,  darkness 
will  remain  during  rotation  about  it.  In 
Fig.  707  if  ab  represents  the  plane  of  the  optic  axes,  the  axis  /  will 
coincide  with  the  optic  normal  (b  axis  of  the  Fresnel  ellipsoid  or  (3 
of  the  indicatrix).  The  angle  at  which  the  axis  H  is  inclined  in  the 
horizontal  plane  may  now  be  read  from  the  outer  ring  (T\,  Fig.  695).  The 
stereographic  net  may  be  turned  through  a  similar  angle  so  that  the  principal 

1  E.  von  Fedorow:  Op.  cit.,  Zeitschr.  f.  Kryst.,  XXIX  (1897-8),  606-610. 


ART.  436]  MEASUREMENT  OF  THE  OPTIC  AXIAL  ANGLE  501 

diameter  coincides  with  the  direction  of  this  axis  (Fig.  707).  Determine  the 
inclination  of  H,  and  indicate  this  position  on  the  stereographic  projection  as  it 
would  appear  rotated  to  the  horizontal  plane.  Every  point  on  the  sphere  will 
describe  a  circle  at  right  angles  to  the  H  axis  and  be  projected  as  a  vertical  small 
circle.  Thus  in  the  figure  a  rotation  of  22°  is  shown,  the  end  of  the  /  axis 
(  =  /3)  appearing  at  c,  and  the  desired  optic  axis  as  OA'  on  a  line  at  right'angles 
to  HH '.  The  plane  of  ab  will  appear  as  the  circle  a'b'  in  the  projection,  every 
point  in  it  lying  22°  distant  on  the  small  circles.  An  arc  may  be  drawn  through 
the  points  so  found  or,  more  simply,  the  curve  may  be  sketched  by  rotating 
the  paper  above  a  Wulff  net.  If  the  inner  glass  circle  of  the  universal  stage 
has  been  rotated,  the  orientation  of  the  optic  axes  with  respect  to  crystallo- 
graphic  directions  may  be  obtained  by  simply  rotating  the  entire  net  through 
the  proper  angle;  for  example,  by  transferring  points  by  means  of  a  trans- 
parent Fedorow  net. 

To  determine  the  location  of  the  other  optic  axis,  rotate  about  J  through 
some  round  number  of  degrees,  and  determine  the  extinction  angle  in  this 
position.  The  extinction  curves  thus  obtained  will  intersect  a'  A '  b'  at  the 
second  optic  axis. 

Determine  graphically  the  positions  of  7  and  a.  The  first  determination, 
on  account  of  the  indistinct  extinction,  is  to  be  regarded  as  approximate,  and 
is  to  be  corrected  by  the  redetermination  of  the  symmetry  planes.  This, 
however,  is  a  simple  process,  since  their  approximate  positions  are  now  known. 

435-  (c)  One  Optic  Axis  makes  an  Angle  of  between  20°  and  55°  with  the 
Normal  to  the  Section,  the  Other  lies  beyond  55°. — The  first  step  in  the 
determination  of  extinction  curves  is  used  for  this  determination.  The  first 
optic  axis  is  rotated  until  it  lies  in  the  plane  at  right  angles  to  the  axis  /. 
In  this  position  the  extinction  angle  is  determined  with  the  stage  in  horizontal 
position,  as  well  as  inclined  to  some  round  number  of  degrees  (corrected  angle). 
By  this  means  are  obtained  a  diameter  and  a  great  circle  in  the  projection, 
and  their  intersection  gives  the  location  of  the  other  optic  axis.  Now  de- 
termine graphically  7,  0,  and  a,  and  verify  by  symmetry  planes. 

436.  (d)  Both  Optic  Axes  are  Inclined  more  than  55°  to  the  Normal  to 
the  Section. — In  this  case  the  inner  glass  stage  is  set  at  o°,  and  the  mineral 
section  is  rotated  about  M  and  H  to  the  point  of  darkness.  The  section  is 
now  rotated  about  /  to  test  whether  the  darkness  remains.  If  it  does  not 
do  so,  it  is  rotated  to  a  different  position  of  darkness  about  M  and  H,  until 
finally,  after  repeated  trials,  it  remains  dark  also  during  the  rotation  about 
/.  In  this  position  the  axis  /  coincides  with  one  of  the  axes  of  the  optical 
ellipsoid,  and  therefore  a  symmetry  plane  lies  at  right  angles  to  this  axis. 

The  amount  of  inclination  of  the  H  axis  to  the  vertical  cross-hair,  that  is, 
its  rotation  about  M  in  the  horizontal  plane,  is  shown  by  rotating  the  stereo- 
graphic  net  to  an  equal  angle  (Fig.  707).  The  pole  of  the  ellipsoid  axis  may 


502  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  436 

now  be  located  on  the  vertical  small  circle  Jc,  by  laying  off  from  J  to  c  a 
distance  equal  to  the  amount  of  the  vertical  inclination  about  the  #axis. 
The  symmetry  plane,  corresponding  to  this  pole,  is  located  directly  by  laying 
off  from  the  line  ab,  along  vertical  small  circles,  angles  equal  to  Jc.  This  gives 
the  great  circle  a'b',  which  is  the  rotated  position  of  ab. 

The  operation  is  now  repeated  for  other  angles  of  H,  to  locate  a  second 
pole  and  a  second  symmetry  plane.  Knowing  two  planes,  the  third  may  be 
constructed  graphically. 

The  position  of  the  symmetry  planes  must  be  verified  now  in  the  usual  way, 
since  the  above  operation  gives  simply  the  approximate  positions.  If  one 
optic  axis  is  not  inclined  over  70°,  its  position  may  best  be  determined  by 
means  of  an  optical  curve.  Its  position  is  at  the  intersection  of  this  curve 
with  the  plane  of  the  optic  axes.  The  other  optic  axis  may  easily  be  deter- 
mined graphically. 

If  both  optic  axes  are  inclined  over  70°  they  may  be  determined  by  the 
method  given  in  Art.  431. 


CHAPTER  XXXVI 

DETERMINATION  OF  OTHER  PROPERTIES  THAN  2V  BY  MEANS 
OF    THE    UNIVERSAL    STAGE 

437.  Opaque  Minerals. — The  universal  stage  offers  a  ready  means  for 
changing  the  angles  at  which  the  incident  light  falls  upon  opaque  minerals, 
thus  aiding  in  their  examination  by  reflecting  the  light  from  their  surfaces. 

438.  Isotropic,  Uniaxial,  or  Biaxial  Character. — Isotropic  crystals  remain 
dark  in  every  position  between  crossed  nicols,  consequently  if  an  isotropic 
section  is  rotated  in  altitude,  for  example  about  the  /  axis  of  the  universal 
stage,  it  will  remain  dark.     This  will  also  occur  in  uniaxial  and  biaxial 
crystals  if  a  symmetry  plane  of  the  optical  ellipsoid  happens  to  lie  at  right 
angles  to  the  /  axis.     A  slight  rotation  about  M,  however,  with  /  in  an  in- 
clined position,  will  definitely  show  whether  or  not  the  crystal  is  isotropic. 

Uniaxial  crystals  may  be  separated  from  biaxial  crystals  by  tilting  the 
mineral  section  until  a  position  is  reached  in  which,  during  a  complete  rota- 
tion about  the  M  axis,  the  stage  will  remain  uniformly  dark.  If  the  crystal 
is  uniaxial,  two  symmetry  planes  will  pass  through  this  point,  consequently 
the  stage  may  be  rotated  about  the  /  and  the  H  axes,  at  right  angles  to  each 
other,  and  darkness  will  remain.  If  the  crystal  is  biaxial,  but  one  plane  of 
symmetry,  the  plane  of  the  optic  axes,  will  pass  through  this  point. 

439.  Positive  or  Negative  Character  of  an  Anisotropic  Mineral.1 — A 

uniaxial  crystal,  placed  between  crossed  nicols  and  with  its  principal  axis  par- 
allel to  the  /  axis  of  the  stage,  will  show  equal  birefringence  (o>)  in  every 
direction  in  the  zone  at  right  angles  to  the  axis,  except  for  such  differences  as 
may  be  caused  in  the  thickness  of  the  section  by  the  rotation.  When  rotated 
about  the  axis  at  right  angles  to  /,  the  birefringence  vvill  gradually  change 
from  o  to  u-e.  To  determine  the  optical  character  of  the  crystal,  it  is  only 
necessary  to  determine  whether  the  ease  of  vibration,  in  the  direction  of  rota- 
tion about  the  /  axis,  is  greater  or  less  than  in  the  direction  at  right  angles 
to  it. 

A  biaxial  crystal  will  show,  in  general,  increasing  birefringence  in  ooth 
directions  of  rotation.  Select  the  section  showing  the  highest  interference 
colors,  that  is,  the  section  nearest  the  plane  of  the  optic  axes.  Determine  in 
it  the  slow  ray  (c).  Tilt  the  section  as  much  as  possible,  and  rotate  it  about 

1  E.  von  Fedorow:  Ein  einf aches  Verfahren  zur  Bestimmung  des  absoluten  optischen 
Zeichens  eines  unregelmassigen  Miner  alkornchens  in  Diinnschlijfen.  Zeitschr.  f.  Kryst., 
XXIV  (1894-5),  603-605. 

503 


504  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  440 

M  until  it  comes  to  the  position  where  the  interference  colors  are  lowest. 
Now  turn  the  section  back  to  the  horizontal  position  and  determine  the  angle 
between  the  axis  of  rotation  and  the  axis  C.  If  this  angle  is  less  than  45°  the 
mineral  is  positive,  if  greater  negative.  The  above  method  is  only  roughly 
approximate.  For  accurate  determinations  it  is  necessary  to  measure  ac- 
curately the  angle  2V. 

As  an  example,  von  Fedorow  gave  a  determination  made  on  a  crystal  of  epi- 
dote.  A  section  nearly  parallel  to  the  plane  of  the  optic  axes  showed  green 
of  the  third  order.  Inclined  about  50°,  and  rotated  about  the  M  axis,  a  posi- 
tion was  found  in  which  the  second  order  blue  appeared.  The  inclination  of 
H  to  the  direction  of  the  slow  ray  was  10°  to  15°,  consequently  the  mineral  was 
negative. 

440.  Maximum  Extinction  Angle. — In  the  determination  of  feldspars, 
pyriboles, 1  and  many  other  minerals,  it  is  necessary  to  determine  the  maxi- 
mum extinction  angle.     Ordinarily  a  search  is  made  through  the  slide,  and 
the  maximum  angle  of  all  those  found  is  considered  the  maximum  angle  of 
the  mineral.    This  value  may  not  always  be  correct,  for  in  a  schistose  rock 
the  crystals  may  lie  more  or  less  parallel,  consequently  the  orientation  may 
be  such  that  the  maximum  angle  cannot  be  obtained.    In  slides  containing 
feldspars,  there  may  be  two  kinds  of  plagioclase  and,  unless  combined  Carls- 
bad and  albite  twinning  occurs  or  some  other  property  aids  in  the  determina- 
tion, one  determines  simply  the  feldspar  having  the  maximum  extinction 
angle.    With  the  universal  stage  it  is  a  simple  matter  to  rotate  a  section  to 
various  positions  and  note  whether  the  angle  increases  or  decreases.    Thus 
the  maximum  angle  may  be  readily  obtained. 

441.  Mean  Refractive  Index  of  a  Mineral. — A  rough  method  for  deter- 
mining the  value  of  (3  follows  from  the  method  of  determining  symmetry 
planes.    It  has  already  been  given.2 

442.  Orientation  of  the  Crystal  Section  with  Reference  to  the  Axes  of 
the  Optical  Ellipsoid. — The  inclination  of  the  mineral  section  to  the  axes  of 
the  optical  ellipsoid  or  to  the  optic  axes,  or  the  inclination  of  the  optic  axes 
with  respect  to  the  section,  may  be  determined  from  the  methods  given  above 
for  locating  the  axes  and  symmetry  planes  of  the  optical  ellipsoid.3 

443.  Determination  of  the  Maximum  Birefringence  of  an   Unknown 
Mineral  from  that  of  One  which  is  Known. — The  method  of  determining 
the  maximum  birefringence  of  a  mineral  is  best  illustrated  by  an  example.4 

1  Family  of  pyroxenes  and  amphiboles.     See  Albert  Johannsen:  Petrographic  terms 
for  field  use.     Jour.    Geol.,   XIX    (1911),   319. 

2  Art.  429. 

3  Art.  429. 

4  E.  von  Fedorow:  Op.  cit.,  Zeitschr.  f.  Kryst.,  XXV  (1895-6),  355-356. 

See  also  W.  Nikitin:  Beitrag  zur  Universalmethode.  Zur  Bestimmung  der  Doppel- 
brechung.  Zeitschr.  f.  Kryst.,  XXXIII  (1900),  133-146. 


ART.  443]     DETERMINATION  OF  OTHER  PROPERTIES  THAN  2V  505 

It  is  required  to  determine,  from  a  section  of  a  quartz  epidosite.  the  thickness 
of  the  slice  and  the  maximum  birefringence  of  the  epidote  from  a  grain  selected 
at  random. 

A  quartz  grain  is  selected  and  rotated  until  its  c  axis  lies  parallel  to  the 
axis  of  the  microscope.  The  angle  of  inclination  of  the  section  is  found  to  be, 
say,  42°.  This  represents  a  true  angle  of  49  1/2°  (Fig.  698)  since  the  light, 
in  this  case,  passes  from  quartz,  with  a  refractive  index  of  1.544,  to  glass  with 
an  index  of  1.74.  The  stage  of  the  microscope  is  now  turned  until  the  /  axis 
of  the  universal  stage  coincides  in  direction  with  the  principal  section  of  one 
of  the  nicols.  In  this  position,  no  matter  how  much  the  crystal  is  rotated, 
it  remains  dark,  since  crystallographic  c  constantly  remains  in  a  plane  per- 
pendicular to  the  direction  of  the  axis  /. 

The  crystal  is  now  rotated  so  that  crystallographic  c  lies  in  the  plane  at 
right  angles  to  the  axis  of  the  microscope.     Since  the  inclination  of  this  axis 
was  49  1/2°,  the  true  angle  must  be  90°— 
49    i/2°=4o   1/2°,  which  corresponds  to   a      \^ 
rotation  of  the   stage  of  35°.     The  quartz 
crystal  now  lies  in  the   position  giving   the 
maximum  birefringence    (e—  u>).     Measuring 
the  amount  of  the  retardation  by  means  of 
a  quartz  or  mica  wedge,  it  is  found  to  be, 
say,  3io,uju  which,  since  quartz  has  a  maxi- 
mum birefringence  of  0.009,  gives  a  thickness 
of  section  of  0.035  mm.,  as  may  be  found  FlG'  ?o8 

from  Fig.  453.     But  the  thickness  of  section 

which  produced  this  color  was  not  the  true  thickness  of  the  section  but  the 
thickness  along  the  inclined  line  Oa  (Fig.  708).  The  true  thickness  Oc 
equals  Oa  -  cos  40  i/2°  =  o.o35  •  cos  40  i/2°  =  o.o26  mm. 

Instead  of  computing  the  values  of  the  cosines  for  each  mineral,  they  may 
be  computed,  once  for  all,  and  shown  graphically,  as  in  Fig.  709.  In  this 
diagram  the  true  thickness  is  shown  by  the  perpendicular  through  the  inter- 
section of  the  true  angle  with  the  curve  representing  the  measured  thickness. 

Coming  now  to  the  second  part  cf  the  problem.  A  grain  of  epidote  is 
rotated  to  determine  the  positions  of  the  optic  axes,  which  may  not  appear 
very  clearly  on  account  of  the  strong  dispersion  of  epidote.  Having  located 
each  of  the  optic  axes  by  means  of  two  angles,  they  are  corrected  for  their 
refractive  indices,  and  are  located  on  a  stereographic  projection  net.  This 
determines  the  plane  of  the  optic  axes,  from  which  may  be  obtained  the  posi- 
tion of  b  and  the  amount  of  rotation,  likewise  corrected,  necessary  to  revolve 
the  stage  in  order  to  bring  it  parallel  to  the  axis  of  the  microscope,  say  43° 
for  the  /  axis  and  x°  for  the  M.  Carrying  out  this  rotation,  the  double 
refraction  is  determined — perhaps  green  of  the  third  order — representing 
a  wave  difference  of  127.5^.  From  the  determination  on  the  quartz  it  was 


506 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  444 


found  that  Oc  =  0.026  mm.  whereby  Oa 


0.026 


7-5=0.035  mm.     Again,  from 


cos  43 

Fig.  453  we  find  that  a  section  0.035  mm-  in  thickness  and  having  a  retarda- 
tion of  127.5^1^  has  a  maximum  birefringence  of  approximately  0.037.  This 
is  the  desired  maximum  birefringence  of  the  epidote. 

,90T 


.02  .03 

True  Thickness 

PIG.  709. — Graphical  solution  of  the  equation,  b  =  c  cos  a,  used  in  reducing  the  measured  thickness  of 

a  section  to  its  true  thickness. 

444.  Graphical  Representation  of  the  Variation  in  the  Double  Refraction 
in  Different  Directions. — The  variation  in  the  double  refraction  in  different 
directions  in  a  crystal,  as  obtained  by  the  rotating  stage,  may  be  shown  in 
stereographic  projection  in  a  manner  similar  to  that  given  by  Schneiderhohn1 
for  the  variation  observed  by  means  of  a  sliding  diaphragm  (Blendenschieber) 
above  the  ocular.  By  moving  the  slide  in  this  ocular  from  the  center  across 
the  field,  parallel  to  one  or  the  other  of  the  nicol  prisms,  only  a  single  inclined 
beam  of  light  will  pass  through  the  section,  the  amount  of  retardation  increas- 
ing as  the  distance  traveled  through  the  slice,  consequently  the  inclination, 

1  H.  Schneiderhohn:  Die  Beobachtung  der  interferenzfarben  schiefer  Strahlenbundel  als 
diagnostisches  Hilfsmittel  bei  mikroskopischen  Miner  aluntersuchungen.  Zeitschr.  f.  Kryst., 
L  (1912),  231-241. 


ART.  444]         DETERMINATION  OF  OTHER  PROPERTIES  THAN  2V         507 


increases. 


While  the  variations  are  too  slight  to  be  detected  with  accuracy  hi 
ordinary  cases  in  the  small  space  permitted  by  the  opening  angle  of  the  ocular, 
the  diagrams  given  by  Schneiderhohn  may  be  studied  with  advantage.  In- 
stead of  showing  lines  of  equal  retardation,  as  was  done  by  Michel-Levy  and 
others,1  Schneiderhohn  indicates  an  increase  or  decrease  by  increasing  or  de- 


Q 

FIG.  710.  PIG.  711.  PIG.  712.  FIG.  713. 

FIGS.  710  to  713. — Graphical  representation  of  the  variation  in  the  strength  of  the  double  refraction 
in  different  directions  in  a  crystal. 

creasing  the  width  of  a  broad  shaded  line.  Thus  Fig.  710  represents  the  in- 
creasing and  decreasing  birefringence  in  zones  along  the  vibration  planes  of  the 
nicols  of  a  section  of  a  uniaxial  mineral  cut  parallel  to  the  optic  axis.  Fig. 
711  shows  an  inclined  uniaxial  crystal,  Fig.  712  a  biaxial  crystal  cut  perpen- 
dicular to  the  acute  bisectrix,  and  Fig.  713  a  random  section  of  a  biaxial 
crystal. 

1  Art.  289. 


CHAPTER  XXXVII 
OPTICAL  ANOMALIES 

445.  The  Cause  of  Optical  Anomalies. — So  long  ago  as  1815,  Brewster 
recognized  the  fact  that  certain  crystals  show  optical  properties  which 
are  not  in  harmony  with  their  physical  characters.  Thus  many  crystals  of 
leucite,  analcite,  garnet,  and  boracite,  which  are  isometric  and  should  there- 
fore appear  isotropic,  show  low  interference  colors  between  crossed  nicols, 
and  may  even  show  a  uniaxial  or  a  biaxial  interference  figure  in  convergent 
light.  In  other  cases  the  same  minerals  show  bands  which  extinguish  in  dif- 
ferent positions.  This  is  well  shown  in  leucite.  Another  anomaly  in  an 
isotropic  substance  is  the  double  refraction  seen  in  stained  glass. 

Again,  crystals  of  the  hexagonal  or  tetragonal  systems  may  show  biaxial 
interference  figures.  This  is  very  common  in  quartz,  eudialyte,  and  nephelite, 
the  former,  in  many  cases,  showing  an  apparent  axial  angle  of  18°,  while 
eudialyte  sometimes  has  one  as  great  as  50°.  Another  anomaly  is  the  sep- 
aration of  basal  sections  of  tetragonal  crystals  into  sectors  of  different  illumi- 
nation. This  is  well  shown  in  apophyllite  and  vesuvianite1  which,  between 
crossed  nicols,  appear  to  be  made  up  of  a  number  of  separate  triangles  joined 
at  their  apices. 

Various  theories  have  been  advanced  to  account  for  these  optical  anomalies, 
and  it  is  probable  that  not  all  cases  are  due  to  the  same  cause.  That  compres- 
sion or  tension  is  able  to  change  the  optical  character  of  a  mineral  was  already 
recognized  by  Brewster,  and  many  experiments  have  been  made  by  subse- 
quent investigators  on  crystal  sections  and  on  colloids. 

The  effect  of  pressure  on  an  isotropic  substance  may  most  easily  be  shown 
by  inserting  a  perfectly  circular  disk  of  soft  gelatine  between  two  object 
glasses  and  placing  it  on  the  stage  of  a  polariscope.  If  the  ocular  tube  is 
lowered  until  it  touches  the  upper  glass,  and  a  little  pressure  is  applied,  a  per- 
fect uniaxial  cross  will  appear.  The  reason  is  not  far  to  seek.  The  gelatine, 
when  not  under  pressure,  was  isotropic,  and  the  light  passed  through  with 
equal  ease  in  every  direction.  Upon  the  application  of  vertical  pressure,  the 
stress  developed  in  this  direction  became  greater  than  in  the  direction  at  right 
angles  to  it,  which,  in  a  circular  disk,  is  radially  equal.  As  a  result,  the  indi- 
catrix  was  changed  from  a  perfect  sphere  to  an  ellipsoid  of  rotation.  If  the 
interference  figure  thus  produced  is  examined  by  the  aid  of  the  gypsum  plate, 

1  See  Klocke,  Neues  Jahrb.  1881  (I),  204-205;  Klein, Neues  Jahrb.,  1884  (I),  253-256. 
References  in  bibliography  at  end  of  chapter. 

508 


ART.  44.5]  OPTICAL  ANOMALIES  509 

it  will  be  found  that  it  is  negative,  which  follows  from  the  fact  that  the  in- 
crease in  pressure  has  increased  the  ease  of  vibration  in  the  same  direction. 
The  indicatrix,  being  the  inverse  of  the  ease  of  vibration  figure,  will  con- 
sequently be  oblate.  If  the  pressure  is  applied  around  the  periphery,  as 
may  be  done  by  surrounding  the  gelatine  disk  by  a  brass  strip  and  drawing 
it  together  by  a  cord,  the  ellipsoid  will  be  prolate  and  the  figure  positive. 

To  show  the  effect  of  pressure  on  mineral  sections,  Bucking  devised  the 
instrument  shown  in  Fig.  714.  A  brass  plate  b  is  clamped  to  the  stage  of  the 
polariscope  so  that  the  opening  o  is  in  the  center.  A  steel  plate  d  is  screwed 
to  b  on  one  side,  and  on  the  other  is  attached  the  sliding  plate  e.  The  latter 
may  be  forced  against  a  crystal,  placed  over  the  opening  o,  by  the  screw  m, 
and  the  amount  of  the  pressure  may  be  measured  by  the  compression  indi- 
cated on  the  frame  r. 


FIG.  714. — Backing's  instrument  for  showing  the  effects  of  pressure  upon  the  interference  figure  of  a 

mineral.     (Fuess.) 

If  a  cube  of  a  uniaxial  crystal  is  placed  between  the  jaws  of  this  instrument, 
so  that  the  optic  axis  lies  at  right  angles  to  the  plate  b,  and  pressure  is  applied, 
a  gradual  opening  of  the  uniaxial  cross  is  seen.  The  instrument  should  be  so 
arranged  on  the  stage  that  the  compression  comes  at  45°  to  the  principal 
sections  of  the  nicols.  By  increasing  the  pressure,  the  hyperbolae  separate  still 
farther,  but  not  in  proportion  to  the  amount  of  pressure,  for  while  a  very 
slight  pressure  will  make  a  uniaxial  crystal  biaxial,  considerable  pressure  is 
necessary  to  increase  the  size  of  the  optic  angle. 

If  a  cube  of  glass  is  placed  in  the  instrument,  the  effect  of  the  lateral 
pressure  produces  an  interference  figure  resembling  that  of  a  uniaxial  crystal 
cut  at  right  angles  to  the  direction  of  its  optic  axis. 

A  biaxial  crystal,  cut  with  its  acute  bisectrix  vertical,  when  compressed  will 
show  an  increase  or  a  decrease  in  the  optic  axial  angle,  depending  upon  the 
direction  of  the  pressure  and  the  optical  character  of  the  mineral.  In  every 
case,  pressure  increases  the  ease  of  vibration  in  the  direction  in  which  it  is 
applied. 

Tension  and  compression,  then,  easily  account  for  the  anomalous  biaxial 
character  of  certain  uniaxial  crystals,  and  the  greater  or  less  size  of  the  optic 
axial  angle  of  those  that  are  biaxial.  One  may  visualize  the  change  produced 
by  imagining  compression  or  tension  exerted  upon  the  optical  ellipsoids. 

To  account  for  the  separation  of  certain  minerals  into  differently  illumi- 
nated fields,  between  crossed  nicols,  Reusch  supposed  that  such  minerals 


510  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  445 

contracted  in  certain  directions  during  the  process  of  crystallization  and, 
upon  solidification,  retained  the  strain  thus  induced.  That  such  is  actually 
the  case  was  shown  by  Klein  and  by  Ben  Saude  who  filled  molds  with  gelatine 
'  and  allowed  them  to  dry  for  two  to  three  days.  It  was  found  that  slices, 
mounted  in  Canada  balsam  to  prevent  further  drying,  showed  a  separation 
into  fields,  as  do  certain  isotropic  minerals,  and  that  these  fields  were  depend- 
ent upon  the  outlines  of  the  molds  used. 

Other  optical  anomalies,  such  as  double  refraction  in  leucite  or  boracite, 
may  be  explained  by  the  fact  that  these  minerals  are  dimorphous,  that  is, 
possess  two  forms.  Above  433°  leucite  is  truly  isometric,  while  below  this 
temperature  it  has  weak  double  refraction.  It  therefore  crystallized  from 
the  igneous  magma  in  the  isometric  system,  and  retained  its  original  form 
upon  cooling. 

Another  cause  for  optical  anomalies  may  be  the  intergrowth  of  lamellae 
in  slightly  different  optical  orientation,  as  in  prehnite,  or  in  lamellae  of  slightly 
different  chemical  composition,  as  in  alum. 

The  abnormal  interference  colors  spoken  of  in  Art.  290,  and  caused  by 
different  retardations  in  rays  of  different  wave  lengths,  are  sometimes  called 
anomalous  interference  colors. 


GENERAL  BIBLIOGRAPHY— OPTICAL  ANOMALIES 

1815.  Sir  David  Brewster:  On  the  optical  properties  of  muriate  of  soda,fluate  of  lime,  and  the 

diamond,  as  exhibited  in  their  action  upon  light.     Trans.  Roy.  Soc.  Edinburgh, 
VIII  (1818),  157-163.     (Read  to  the  Society  Nov.  24,  1815.) 

1816.  Idem:  On  the  effects  of  compression  and  dilation  in  altering  the  polarizing  structure  of 

doubly  refracting  crystals.     Ibidem.,  281-6. 

1821.  Idem:  A  new  primitiie  form  of  boracite.     Edinburgh  Phil.  Jour.,  V  (1821),  217. 

1822.  Idem:  On  a  new  species  of  double  refraction,  accompanying  a  remarkable  structure 

in  the  mineral  called  analcime.     (Read  Jan.  7,  1822).     Trans.  Roy.  Soc.  Edin- 
burgh, X  (1826),  187-194. 
1833.  Idem:  Observation  relatiie  to  the  structure  and  origin  of  the  diamond.     Abstract  of  a 

paper  in  Proc.  Geol.  Soc.,  Phil.  Mag.  3d  ser.,  Ill  (1833),  219-220. 

1841.  J.  B.  Biot:  Sur  la  polarisation  lamellaire.     Comptes  Rendus,  XII  (1841),  967-979. 
Idem:  Analyse   experimentale  des  phenomenes  de  polarisation  produits  par  les  corps 
cristallises  en  lertu  d'un  action  non  moleculaire.  i  partie.  Ibidem.,  XIII  (1841), 
155-162. 

Idem:  Sur  la  polarisation  lamellaire.     Ibidem.,  XIII  (1841),  391-397. 
Idem:  Particularity  relatives    aux   cristaux  d'apophyllite.     Ibidem.,  XIII    (1841), 

839-840. 
Idem:  Memoire  sur  la  polarisation  lamellaire.      M6m.  Acad.  France,  XVIII  (1841), 

539-725'     (Contains  the  above  papers  in  full.) 

1841.  F.  E.  Neumann:  Die  Gesetze  der  Doppelbrechung  des  Lichts  in  comprimirten  oder 
ungleichformig  erwdrmten  unkrystallinischen  Korpern.  Pogg.  Ann.,  LIV  (1841), 
449-476. 

1851.  Wertheim:  Note  sur  la  double  refraction  arlificiellement  produite  dans  des  cristaux  du 
systems  reguliere.  Comptes  Rendus,  II  (1851),  576-579. 


ART.  445]  OPTICAL  ANOMALIES  511 

1855.  H.    Marbach:  Ueber  die   optischen   Eigenschaften   einiger   Krystalle   des   tesseralen 

Systems.     Pogg.  Ann.,  XCIV  (1855),  412-426. 
1857.  Volger:  Monographic  des  Boracits.     Hannover,  1857.* 
1857.  A.  des  Cloizeaux:  De  Vemploi  des  proprietes  optiques  birefringentes  en  mineralogie. 

Ann.  d.  Mines,  XI  (1857),  261-342. 
Idem :   Sur  Vemploi   des   proprietes   optiques  birefringentes  pour  la  determination  des 

epeces  cristallisees.     Ibidem,  XIV  (1858),  339-420. 
1859.  Friedrich  Pfaff:   Versuche  iiber  den  Einfluss  des  Drucks  auf  die  optischen  Eigcn- 

schaflen  doppeltbrechender  Krystalle.     Pogg.  Ann.,  CVI1  (1859),  333-338;  CVIII 

(1859),  598-601. 
1867.  E.    Reusch:  Ueber  die  sogenannte  Lamellar  polarisation  des  Alauns.     Pogg.  Ann., 

CXXXII  (1867),  618-622. 
1871.  Aristides  Brezina:  Die  Kryslallform  des  unterschu>efelsauren  Blei  PbSzOt.^aq  und  das 

Gesttz  der   Trigonoeder  an  circularpolarisirenden   Krystalle.     Sitzb.  Akad.  Wiss. 

Wien.,LXiV  (I),  1871,  289-328. 
1875.  J.  Hirschwald:  Zur  Kritik  des  Leudtsy  stems.     T.M.P.M.,  1875,  227-250. 

1875.  A.  Wichmann:  Note  in  Zeitschr.  d.  deutsch.  geol.  Gesell.,  XXVII  (1875),  749-751- 

1876.  Arthur  Wichmann:  Ueber  doppelbrechende  Granaten.     Pogg.  Ann.,  CLVII  (1876), 

282-290. 

1876.  Er.  Mallard:  Explication  des  phenomenes  optiques  anomaux  que  presentent  un  grand 

nombre  de  substances  cristallisees.     Ann.  d.  Mines,  X  (1876),  60-196. 

1877.  P-  Groth:  Ueber  anomale  optische  Erscheinungen  an  Krystallen.     Zeitschr.  f.  Kryst., 

I  (1877),  309-320.     Review  of  preceding. 

1878.  Er.  Mallard:  Note  in  Bull.  Soc.  Min.  France,  I  (1878),  107-110. 

1878.  A.  von  Lasaulx:   Ueber  das  optische  Verhalten  und  die  Krystallform  des  Tridymites. 

Zeitschr.  f.  Kryst.,  II  (1878),  253-274. 

1879.  H.  Baumhauer:   Ueber  den  Per ou'S kit.     Zeitschr.  f.  Kryst.,  IV  (1879),  187-300. 

1879.  Ed.  Jannettaz:  Sur  les  colorations  du  diamant  dans  la  lumiere  polarisee.     Bull.  Soc. 
Min.  France,  II  (1879),  124-131. 

1880.  Er.  Mallard:  Sur  les  proprietes  optiques  des  melanges  de  substances  isomorphes  et  sur 

les  anomalies  optiques  des  cristaux.     Bull.  Soc.  Min.  France,  III  (1880),  3-20. 
1880.  Ed.  Jannettaz:  Reponse  a  la  note  precedente  de  M.  Mallard.     Ibidem,  20-24. 
1880.  Johann   Rumpf:  Ueber  den   Krystallbau  des   Apophyllits.     T.M.P.M.,   II    (1880), 

369-391. 
1880.  Friedrich  Becke:  Ueber  die  Zwillingsbildung  und  die  optischen  Eigenschaften  des 

Chabasit.    T.M.P.M.,  II  (1880),  391-418. 
1880.  F.   Klocke:  Ueber  Doppelbrechung  regiddrer  Krystalle.     Neues   Jahrb.,  1880    (I), 

53-88. 
1880.  Idem:  Ueber  ein  optisch  anomales  Verhaiten  des  unterschwefelsauren  Blei.     Xeues 

Jahrb.,  1880  (II),  97-99. 
1880.  H.   Bucking:  Ueber  durch  Druck  hervorgerufene  optische  Anomaiien.     Zeitschr.   d. 

deutsch.  geol.  Gesell.,  XXXII  (1880),  199-202. 
1880.  A.  de  Schulten:  Sur  la  reproduction  artificielle  de  V  Analcime.     Bull.  Soc.  Min.  France, 

III  (1880),  150-153- 

1880.  C.  Klein:  Ueber  den  Boracit.     Xeues  Jahrb.,  1880  (II),  209-250,  especially  209-217. 

1881.  Er.  Mallard:  Sur  la  theorie  des  phenomenes  produits  par  des  croisements  de  lames 

cristallines  et  par  des  melanges  de  corps  isomorphes.     Bull.  Soc.  Min.  France,  IV 

(1881),  71-79- 
1 88 1.  F.  Klocke:  Ueber  ein  optisch  analoges  Verhalten  einiger  doppeltbrechender  regular er 

mil  optisch  zu'eiaxig  erscheinenden  letragonalen  Krystallen.     Neues  Jahrb.,  1881  (I), 

204-205. 
1881.  A.  Arzruni  und  S.  Kock:  Ueber  den  Analcim.     Zeitschr.  f.  Kryst.,  V  (1881),  483-489. 


512  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  445 

1881.  C.  Klein:  Zur  Frage  iiber  das  Kryslallsyslem  des  Boracit.     Neues  Jahrb.,  1881  (I), 

239-256. 
1 88 1.  F.  Klocke:  Ueber  einige  optische  Eigenschaften  optisch  anomaler  Krystalle  und  deren 

Nachahmung  durch  gespannte  und  gepresste  Colloide.     Neues  Jahrb.,  1881   (II), 

249-268. 

1 88 1.  Emile    Bertrand:  Sur  les  cristaux    pseudo-cubiques .     Bull.  Soc.  Min.    France,   IV 

(1881)  237-241. 

1882.  Er.  Mallard:  De  V action  de  la  chaleur  sur  les  cristaux  de  boracit.     Bull.  Soc.  Min. 

France,  V  (1882),  144-159. 

1882.  Alfredo  Ben-Saude:  Uber  den  Analcim.     Neues  Jahrb.,  1882  (I),  41-74. 
1882.  Idem:  Ueber  den  Perowskit.     Preissschrift,  Gottingen,  1882.* 

1882.  Emile  Bertrand:  Sur  les  differences  entre  les  proprietes  optiques  des  corps  cristallises 

birejringents  et  celles  que  peuvent  presenter  les  corps  monorefringents,  apres  qu'ils 
ont  ete  modifies  par  des  retraits,  compressions,  dilatations  ou  toute  autre  cause.  Bull. 
Soc.  Min.  France,  V  (1882),  3-7. 

1883.  G.     vom    Rath:  Ueber    ungewdhnliche    Leucitkrystalle.     Verh.    naturhist.    Verein. 

Bonn,  1883,  II5~II5  °f  Sitzb. 
1883.  H.    Bucking:  Ueber   den   Einfluss   eines   messbaren   Druckes    auf  doppeltbrechende 

Mineralien.    Zeitschr.  f.  Kryst.,  VII  (1883),  555-569. 
1883.  C.  Klein:  Optische  Studien  am  Granat.     Neues  Jahrb.,  1883  (I),  87-163,  especially 

158-163.     Reprinted,  with  alterations  and  additions  by  the  author  from  Nach- 

richten  Gesel.  Wiss.  Gottingen,  1882. 
1883.  R.  Brauns:  Ueber  die  Ursache  der  anomalen  Doppelbrechung  einiger  regular  krystal- 

lisirender  Salze.     Neues  Jahrb.,  1883  (II),  102-111. 

1883.  A.   Ben-Saude:  Anomalias  opticas  de  crystaes  tesseraes.     Segunda  Parte.     Contri- 

buiqoes  para  a  theoria  das  anomalias  opticas.  Jornal  de  Sciencias  mathematicas, 
physicas  e  naturaes.  Lisboa,  XXXVI  (1883),  31  et  seq. * 

Idem.     German   translation  of    above.     Beitrag    zu   einer    Theorie   der   optischen 
Anomalien  der  regular  en  Krystalle.    Lisbon,  1894.* 

1884.  S.  L.  Penfield:  Ueber  Erwdrmungsversuche  an  Leucit  und  anderen  Mineralien.     Neues 

Jahrb.,  1884  (II),  224. 

1884.  Gustav  Tschermak:  Lehrbuch  der  Miner alogie,  Wein,  1884,  196,  473.* 
Idem:  Ibidem,  2  Aufl.,  1885,  196-200. 
Idem:  Ibidem,  3  Aufl.,  1888,  200-204. 
Idem:  Ibidem,  6  Aufl.,  1895,  248-250. 
1884.  Wilhelm  Klein:  Beitrdge  zur  Kenntniss  der  optischen  Aenderungen  in  Krystallen  unter 

dem  Einflusse  der  Erwarmung.     Zeitschr.  f.  Kryst.,  IX  (1884),  38-72. 
1884.  C.  Klein:  Beitrdge  zur  Kenntniss  des  Boracit.     Neues  Jahrb.,  1884  (I),  235-245. 
1884.  Idem:  Perowskit  von  Pfitsch  in  Tirol.     Ibidem,  245-250. 
1884.  Idem:  Analcim  von  Table  Mountain  bei  Golden,  Colorado.     Ibidem. 
1884.  Idem:  Apophyllit  von  Table  Mountain,  Golden,  Colorado,  von  den  Fdroer  Inseln  und 

von  Guanajuato,  Mexico.     Ibidem,  253-256. 

1884.  A.  Merian:  Beobachtung  am  Tridymit.     Neues  Jahrb.,  1884  (I),  193-195. 
1884.  C.  Klein:  Ueber  das  Krystallsystem  des  Leucit  und  den  Einfluss  der  Wdrme  auf  seine 

optischen  Eigenschaften.     Nachr.  Gesell.  Wiss.  Gottingen,  1884,  129-136. 
1884.  C.  Klein:  Oplische  Studien  am  Leucit.     Ibidem,  421-472. 
1884.  C.  Klein:  Ueber  den  Einfluss  der  Wdrme  auf  die  optischen  Eigenschaften  von  Aragonit 

und  Leucit.     Neues  Jahrb.,  1884  (II),  49-50. 

1884.  C.    Doelter:  Erhitzungversuche    an     Vesuvian,    Apatit,    Turmalin.     Neues    Jahrb., 

1884  (II),  217-221. 

1885.  R.  Brauns:  Einige  Beobachtungen  und  Bemerkungen  zur  Beurtheilung  optisch  anoma- 

ler Krystalle.     Neues  Jahrb.,  1885  (I),  96-118. 


ART.  445]  OPTICAL  ANOMALIES  513 

1885.  C.  Klein:  Ueber  die  Ursache  optischer  Anomalien  in  einigen  besonderen  Fallen.     Xeues 

Jahrb.,  1885  (II),  237-239. 

1886.  Er.  Mallard:  Snr  les  hypotheses    diverses    propose.es  pour   expliqutr   les    anomalies 

optiques  des  cristaux.     Bull.  Soc.  Min.  France,  IX  (1886),  54-74. 

1887.  R.  Brauns:  Zur  Frage  der  optischen  Anomalien.     Neues  Jahrb.,  1887  (I),  47-57. 
1887.  Carl  Klein:  Optische  Untersuchungen  zweier   Granatwrkommen  vom  Harz.     Neues 

Jahrb.,  1887  (I),  200-201. 
1887.  Idem:  Beleuchtung  und  Zuriickweisung  einiger  gegen  die  Lehre  von  den  optischen 

Anomalien  erhobenen  Einwendungen.     Neues  Jahib.,  1887  (I),  223-246. 
1887.  R.  Brauns:  Was  unssen  wir  iiber  die  Ursachen  der  optischen  Anomalien?     Verhandl. 

Naturhist.     Vereins,  Bonn,  XLIV  (1887),  510-537. 

1889.  Friedrich  Pockels:  Ueber  den  Einfluss  elaslischer  Deformalionen,  speciell  einseitigen 

Dr ucks,   anj  das  Optische    Verhalten  krystallinischer  Korptr.     (Gives   historical 
summary.)     Wiedem.  Ann.,  XXXVII  (1889),  144-172,  269-305. 

1890.  Idem:  Uebtr  die  durch  einseitigen  Druck  heroorgerufene  Doppelbrechung    regular er 

Krystalle,  speciell   von  Steinsalz  und   Syhin.     Wiedem.  Ann.,  XXXIX   (1890), 
440-469. 

1891.  A.  Karnojitsky:  Einige    Belrachtungen    iiber   die    mb'gliche    Ursache   der    oplischen 

Anomalien  in  den  Kryslallen.     Zeitschr.  f.  Kryst.,  XIX  (1891),  571-592. 

1891.  R.  Brauns:  Die  optischen  Anomalien  der  Krystalle.     Gekronte  Preisschrift,  Heraus- 

gegeben  von  der  Fiirstl.  Jablonski'schen  Gesell.  zu  Leipzig,  1891.     (This  is  espe- 
cially noteworthy  and  contains  a  complete  bibliography  to  1891.)* 

1892.  C.  Klein:  Ueber  das  Kryslallsystem  des  Apophyllils  und  den  Einfluss  des  Drucks  und 

der  Warme  auf  seine  optischen  Eigenschaften.     Neues  Jahrb.,  1892  (II),  165-231. 

1893.  Ff-  Pockels:  Ueber  die  Aenderung  des  optischen  Verhaltens  von  Alaun  und  Beryll 

durch  einseitigen  Druck.     Neues  Jahrb.,  B.  B.  VIII  (1893),  217-268. 
1893.  R.    Brauns:  Review  of   A.   Karnojitsky:  Einige  Belrachtungen  iiber  die  mogliche 
Ursache  der  optischen  Anomalien  in  den  Krystallen.     Neues   Jahrb.,  1893    (I), 

456-457- 

1893.  F.  Zirkel:  Lehrbuch  der  Petrographie,  Leipzig,  1893,  I,  362. 

1894.  A.  Ben-Saude:  Beitrag  zu  einer  Theorie  der  optischen  Anomalien  der  regularen  Krys- 

talle, Lissabon,  1894.* 

1895.  C-  Klein:  Beitrage  zur  Kenntniss  des  Granats  in  optischer  Hinsicht.     Neues  Jahrb., 

1895  TO,  68-106. 
1895.  Idem:  Optische  Studien  am  Vesuvian.     Neues  Jahrb.,  1895  (II),  106-119. 

1895.  R-  Brauns:  Einige  Bemerkungen  zu  dem  von  Herrn  Ben-Saude  gegebenen  Beitrag  zu 

einer  Theorie  der  optischen  Anomalien  der  regularen  Krystalle.     Neues  Jahrb., 
1895  TO,  i33-!43. 

1896.  A.   Ben-Saude:  Die    -wahrscheinlichen   Ursachen  der  anomalen  Doppelbrechung  der 

Krystalle.    Lissabon,  1896.* 

1897.  C.   Klein:  Ueber  Leudt  und  Analcim  und  ihre  gegenseitigen  Beziehungen.     Sitzb. 

Akad.  Wiss.  Berlin,  1897,  290-354. 
Idem:  Same  title.     Neues  Jahrb.,  B.  B.,  XI  (1897-98),  475-553. 

1898.  Reinhard  Brauns:  Ueber  Polymorphic  und  die  optischen  Anomalien  von  chlor-  und 

bromsaurem  Natron.     Neues  Jahrb.,  1898  (I),  40-59. 

1898.  E.  von  Fedorow:  Ueber  eine  besondere  Art  der  optischen  Anomalien  und  der  Sand- 
uhrslructur.     Zeitschr.  f.  Kryst.,  XXX  (1898),  68-70. 

1898.  F.  Wallerant:  Theorie  des  anomalies  optiques,  de  Visomorphisme  et  du  polymorphisme 

deduite  des  theories  de  MM.  Mallard  et  Sohncke.     Bull.  Soc."  Min.  France,  XXI 
(1898),  188-256. 

1899.  E.  S.  Dana:  A  Textbook  o1  Mineralogy.    New  York,  1899,  228-231. 

33 


514  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  445 

1900.  E.  von  Fedorow:  Constatirung  der  optischen  Anomalien  in  Plagioklasen.     Zeitschr. 

f.  Kryst.,  XXXI  (1898-9),  579-582. 

1901.  O.    Miigge:  Krystallographische    Untersuchungen   iiber   die    Umlagerungen   und   die 

Struclur  einiger  mimetischer  Krystalle.     Neues  Jahrb.,  B.  B.,  XIV  (1901),  246-318. 

1904.  Rosenbusch-Wiilfing:  Mikroskopische  Physiographic,  I-i,  Stuttgart,  4  Aufl.,  1904, 

356-359. 

1905.  P.  Groth:  Physikalische  Krystallographie.    Leipzig,  4  Aufl.,  1905,  234-236. 


CHAPTER  XXXVIII 


DETERMINATION  OF  SPECIFIC  GRAVITY 

446.  Specific  Gravity. — The  specific  gravity  or  density  of  a  substance  is 
the  ratio  of.  its  weight  in  air  to  its  weight  in  water  at  4°  C.  (39.2°  F.).     In 
other  words,  it  is  the  ratio  of  the  weight  of  any  fragment  of  a  substance  to  the 
weight  of  an  equal  amount  of  water.     The  specific  gravity  of  a  mineral, 
provided  it  is  pure  and  free  from  inclusions  of  solids,  liquids,  or  gases,  is  a 
constant  quantity.     In  isomorphous  series,  or  in  minerals  whose  chemical 
composition  differs  in  different  speci- 
mens, there  is,   however,  a  variation, 

and  this  serves  as  a  means  of  separa- 
tion. 

The  determination  of  specific  grav- 
ity properly  belongs  to  the  province 
of  mineralogy  and  not  to  petrography, 
for  which  reason  the  usual  methods 
will  be  little  more  than  mentioned 
here.  For  more  detailed  descriptions 
the  student  is  referred  to  the  stand- 
ard works  on  mineralogy.1 

447.  Hydrostatic    Balance. — The 

mineral,  after  examination  under  the 

microscope  for  impurities,  is  weighed 

in  air  (w)  and  then  in  water  (wf).  •  The 

difference  between  these  two  weights 

represents    the   weight    of   an   equal 

amount  of  water  (w— wf).    Therefore  the  specific  gravity  (G)  is  represented 

by  the  equation 


FIG.  715. — Specific     gravity     balance. 
Scientific  Co.) 


(Central 


G  = 


w 


w  —  w 

The  usual  form  of  hydrostatic  balance  is  shown  in  Fig.  715.  It  differs 
from  an  ordinary  balance  only  in  having  one  pan  suspended  by  a  shorter 
wire,  and  in  having  beneath  it  a  hook  to  which  is  attached  a  thin  wire  with  or 

1  See  also  V.  Goldschmidt:  Verhandl.  k.  k.  Geol.  Reichsanst.     Wien,  1886,  439.* 
Idem:  Bestimmung  des  specifischen  GewicJttes  von  Mineralien.     Ann.  d.  k.  k.  naturhist. 
Hofmuseum.,  I  (1886),  127-134. 

515 


516 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  448 


without  another  pan.  For  the  determination  of  the  weight  in  air  the  upper 
pan  is  used,  the  lower  one  being  immersed  in  a  beaker  of  water.  For  the 
determination  of  the  weight  in  water,  the  mineral  is  transferred  to  the  lower 
pan,  if  one  is  present,  or  is  attached  to  the  wire.1 

A  very  convenient  balance  which  gives  at  once  the  value  of  the  specific 
gravity  from  the  reading  on  a  graduated  arm,  was  designed  by  Rogers2  (Fig. 
716). 

If  the  mineral  fragment  is  too  small  to  be  thus  determined,  or  if  it  is  in  a 
powdered  state,  its  specific  gravity  may  be  obtained,  as  suggested  by  Penfield,3 
by  placing  it  in  a  small  glass  tube,  closed  at  one  end,  and  having  a  platinum 
wire  at  the  other  by  which  to  suspend  it.  The  fragments  are  first  weighed 
dry.  They  are  then  boiled  in  water  to  remove  all  air,  are  transferred  to  the 


FIG.   716. — Roger's  specific  gravity  balance. 

tube,  which  is  suspended  from  a  balance,  and  are  weighed  immersed  in  water. 
The  weight  of  the  tube  in  water  without  the  mineral  is  subtracted  from  the 
former  weight  to  give  the  weight  in  water.  The  specific  gravity  is  found  by 
the  same  formula  as  before. 

448.  Jolly  Balance. — In  the  Jolly4  balance  (Fig.  717)  the  specific  gravity 
is  determined  by  noting  the  amount  of  lengthening  of  a  spring  when  the  min- 
eral is  placed  in  the  upper  pan  in  air  (w) ,  and  the  amount  when  it  is  in  the 
lower  pan  and  immersed  in  water  (w'}.  Here  also 

w 

(jr  =  ~      — ,« 

W  —  W 

1  See  Axel  Gadolin:  Eine  einfache  Methode  zur  Bestimmung  des  spezifischen  Gewichtes 
der  Mineralien.     Pogg.  Ann.,  CVI  (1859),  213-225. 

G.  Tschermak:  Ein  einf aches  Instrument  zur  Bestimmung  der  Dichte  der  Miner  alien- 
zugleich  fur  anraherndc  Qiiantitatsbestimmung,  bei  chemischen  Versuchen  brauchbar.  Sitzb, 
Akad.  Wiss.  Wien,  XLVII  (1863),  294-301. 

Franz  Toula:  Hydrostatische  Schnellwage.     T.  M.  P.  M.,     XXVI  (1907),  233-237. 

2  Austin  ¥.  Rogers:  A  new  specific  gravity  balance.     Science,  XXXIV  (1911),  58-60. 

3  S.  L.  Penfield:  Ueber  einige  Verbesserungen  der  Methoden  zur  Trennung  von  Miner- 
alien  mil  hohem  specifischen  Gewicht.     Zeitschr.  f.  Kryst.,  XXVI  (1896),  134-137. 

4  P.  Jolly:  Eine  Federwage  zu  exacten  Wdgungen.     Sitzb.  Akad.  Wiss.  Miinchen,  1864 
(I)  162-166. 


ART.  450] 


DETERMINATION  OF  SPECIFIC  GRAVITY 


517 


An  improved  form  of  Jolly  balance  was  designed  by  Linebarger,1  and  another 
by  Kraus.2 

449.  Pycnometer  for  Determining  the  Specific  Gravity  of  Powders.— 

For  determining  the  specific  gravity  of  small  fragments 
or  of  powders,  the  mineral  may  first  be  weighed  dry  in 
air  (w),  then  placed  in  a  vessel — called  a  pycnometer3 
(Fig.  718) — previously  weighed  full 
of  water  (w'\  the  air  excluded,  the 
water  brought  to  the  same  level  as 
before,  and  the  whole  weighed  (w"). 


G  = 


w+w'-w" 


45o.  Smeeth's  Method  for 
Mineral  Powders  (1888).— Smeeth4  ^cenJaTs^dfic  CoT 
determined  the  specific  gravity  of 
mineral  powders  by  first  heating  a  small  amount  of 
vaseline  in  a  watch  crystal  to  remove  bubbles.  After 
cooling,  the  glass  and  vaseline  were  weighed  in  water 
(w')  by  suspending  them  by  a  fine  wire.  The  watch 
crystal  was  now  taken  out,  the  water  poured  off,  and 
any  remaining  drops  carefully  removed  by  means  of 
filter  paper.  After  heating  the  vaseline  again,  a 
weighed  amount  (w)  of  the  powdered  mineral  was  scat- 
tered over  the  surface  to  which  it  adhered.  The  whole 
was  weighed  in  water,  after  cooling  (w"}. 

w 


The  result  is  entirely  independent  of  the  specific  gravity 
of  the  vaseline. 

FIG.    717. — Jolly  balance  ,  ...  .     .  .         ,    , 

(Central  Scientific  Co.)  C-  E-  Linebarger:  A  new  form  of  the  spiral  spring  balance. 

Physical  Review,  XI  (1900),  iio-m. 

2  Edward  H.  Kraus:  A  new  Jolly  balance.     Amer.  Jour.  Sci.,  XXXI  (1911),  561-563. 
Idem:    Eine   neue    Jolly'sche    Fedencage   zur    Bestimmung   des  spezifischen  Gewichts. 

German  translation  of  preceding.     Centralbl.  f.  Min.,  etc.,  1911,  366-368. 

3  James    P.    Joule    and   Lyon    Play  fair:  Researches    on    atomic    volume    and    specific 
gravity.  Jour.  Chem.  Soc.  London,  I  (1849),  123. 

Earl  of  Berkeley:  On  an  accurate  method  of  determining  the  densities  of  solids. 
Mineralog.  Mag.,  XI  (1895),  64-68. 

W.  Leick:  Ueber  specifische  Gewichtsbestimmung.  Mittheil.  naturwiss.  Ver.  Xeu 
Vorpommern  und  Riigen.  XXVII  (1895).* 

4  \V.  F.  Smeeth:  On  a  method  of  determining  the  specific  gravity   of  substances   in   the 
form  of  powder.     (Communicated  Feb.    14,   1888.)     Proc.  Roy.  Dublin    Soc.,  VI  (1888- 

1890),  61-62. 


518  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  451 

451.  Specific  Gravity  of  Porous  Substances. — In  the  determination  of 
the  specific  gravity  of  porous  substances,  such  as  pumice,  chalk,  etc.,  two 
values  should  be  obtained.     First  the  specific  gravity  of  the  mineral  with  its 
included  air  spaces,  and  second  the  specific  gravity  of  the  material  itself.     The 
former  value  may  be  determined  by  coating  the  mineral  with  a  thin,  and 
negligible,  coating  of  wax  or  varnish,  and  proceeding  as  above.     The  second 
value  is  determined  on  the  mineral  powder. 

452.  Specific  Gravity  of  Substances  Soluble  in  Water. — If  the  substance 
under  examination  is  soluble  in  water,1  its  specific  gravity  with  reference  to 
some  other  fluid,  such  as  absolute  alcohol,  should  be  determined.     The  result- 
ing value  should  be  multiplied  by  the  specific  gravity  of  the  fluid  used. 
Linck  covered  the  mineral  with  a  very  thin  coating  of  paraffine,  prepared  by 
dissolving  a  small  amount  in  much  ether. 

453.  Determination  of  Specific  Gravity  by  Heavy  Fluids. — Instead  of 
determining  the  specific  gravity  of  a  mineral  by  weighing  it  in  water,  the  deter- 
mination may  be  made  by  immersing  it  in  a  liquid  of  known  density  and 
noting  whether  it  sinks  or  floats.     For  such  determinations  a  series  of  fluids 
of  different  specific  gravities,  or  two  fluids  of  widely  different  specific  gravities, 
may  be  used.2    In  the  latter  case  the  mineral  to  be  determined  is  first  placed 
in  the  heavier  solution.     If  it  floats,  the  lighter  fluid  is  slowly  added,  with 
constant  and  thorough  stirring,  until  the  mineral  begins  to  show  signs  of 
sinking.     The  lighter  fluid  is  now  added  more  slowly  until  a  point  is  reached 
at  which  the  mineral  remains  stationary  for  a  short  time  at  any  depth  at 
which  it  may  be  placed.     That  the  point  of  equal  density  is  being  approached 
may  te  seen  by  the  fact  that,  after  stirring,  the  movements  of  the  mineral 
particles  are  more  sluggish,  while  flakes  stand  on  their  edges,  and  laths  on 
their  ends.     When  the  specific  gravity  of  the  mineral  and  the  fluid  are  equal, 
it  is  only  necessary  to  determine  that  of  the  latter. 

Various  fluids  have  been  suggested  for  the  determination  of  density,  but 
these  so-called  heavy  solutions  have  been  used  more  frequently  for  the 
mechanical  separation  of  the  different  components  of  a  composite  rock  than 
for  the  determination  of  specific  gravity. 

The  first  use  of  a  heavy  solution  for  the  determination  of  density  is 
ascribed  by  Kalkowsky3  to  Scheibler  who  used,  in  1861,  sodium  meta- 

1  J.  Linck:    Beitrag  zur  Kenntniss  der  Sulfate  von  Tierraamarilla  bei  Copiapo  in  Chile. 
Zeitschr.  f.  Kryst,  XV  (1888),  1-28,  especially  9. 

J.  W.  Retgers:  Die  Bestimmung  des  spezifischen  Gewichls  von  in  Wasser  loslichen  Salzen, 
Zeitschr.  physikalische  Chemie,  III  (1889),  289-315;  IV  (1889),  189-205;  XI  (1893), 

328-344- 

2  See  caution  in  regard  to  the  use  of  heavy  solutions,  Art.  497. 

3  Review  by  Ernst  Kalkowsky  of  A.  Karpinskij :    Petrographische  Notizen  (Iswestija 
des  geol.  Comites,  III,  No.  8,  263-280),  St.  Petersburg,  1884,  in  Neues  Jahrb.,i886  (I), 
263-264.     In  this  work,  written  in  the  Russian  language,  Karpinskij  gives  a  history  of  the 
discovery  and  use  of  heavy  fluids  for  mechanical  separation  of  rock  constituents.     In  the 
review  the  original  reference  to  Scheibler's  work,  if  published,  is  not  given. 


ART.  454]  DETERMINATION  OF  SPECIFIC  GRAVITY  519 

tungstate  with  a  specific  gravity  of  3.02.  According  to  the  same  authority 
Marignac,  in  1862,  used  a  solution  of  sodium  silico-tungstate  (4Na2OSiO2- 
1 2 WO 3 7 H2O)  with  a  specific  gravity  of  3.05.  In  the  same  year  Schaffgotsch1 
used  an  aqueous  solution  of  acid  mercuric  nitrate.  Into  this  solution  the 
mineral  was  placed.  If  it  floated,  dilute  nitric  acid  was  added  until  the 
mineral  slowly  sank,  whereupon  a  glass  rod  was  dipped  into  the  concentrated 
solution  of  acid  mercuric  nitrate  and  placed  in  the  test  glass  enough  times 
to  cause  the  mineral  to  be  suspended.  The  temperature  of  the  solution 
was  now  raised  to  17  1/2°  C.  by  warming  the  beaker  with  the  hand,  and  the 
specific  gravity  of  the  solution  determined.  On  account  of  the  acid  character 
of  the  solution  it  acts  upon  many  minerals,  and  was  but  little  used.  It  can- 
not be  diluted  with  water  on  account  of  the  precipitation  of  a  basic  salt. 

454.  Sonstadt  (or  Thoulet)  Solution  (1874,  1877).— The  so-called  Thoulet 
solution,  with  a  maximum  specific  gravity  of  3.196,  is  an  aqueous  solution  of 
potassium  mercuric  iodide.  It  was  first  described  by  Sonstadt,2  in  1874, 
and  his  name  should  be  given  to  it.  He  used  it  for  the  determination  of  the 
specific  gravities  of  alkali  salts,  and  prepared  it  by  making  a  saturated  solu- 
tion, at  room  temperature,  of  iodide  of  potassium,  into  which  as  much 
mercuric  iodide  was  stirred  as  would  dissolve 

Though  the  Sonstadt  solution  was  used  by  Church3  in  1877,  it  did  not 
become  generally  known  until  Thoulet4  published  his  experiments  in  1878-9. 
It  became  still  more  widely  known  after  Goldschmidt5  published  his  careful 
investigations  of  the  properties  of  the  solution  in  1881. 

To  prepare  the  solution,  80  c.c.  of  cold  distilled  water  are  taken,  and  in  it 
270  grm.  of  mercuric  iodide  (HgI2)  and  230  grm.  of  potassium  iodide  (KI) 
are  dissolved  by  stirring.  The  solution  is  placed  in  a  porcelain  evaporating 
dish  on  a  water-,  not  sand-bath,  and  is  evaporated  until  a  crystalline  film 
forms  on  the  surface,  or  until  a  crystal  of  tourmaline  (G  =  3.i)  or  fluorite 
(G  =  3.i8)  floats.  Upon  cooling,  the  solution  contracts,  and  the  specific 
gravity  rises  to  its  maximum  of  3.196.  Needles  of  hydrous  potassium 
mercuric  iodide  may  crystallize  out  upon  cooling,  but  if  sufficient  liquid  has 
been  prepared,  the  clear  portion  may  be  decanted.  Should  it  be  necessary* 
to  use  all  of  the  fluid,  a,  few  drops  of  water  added  will  cause  the  crystals  to  be 

1  F.    G.   Schaffgotsch:    Ermittelung  des  Eigengewichts  fester  Ko'rper  durch  Schweben. 
Pogg.  Ann.,  CXVI  (1862),  279-289. 

2  E.  Sonstadt:    Note  on  a  new  method  of  taking  specific  gravities,  adapted  for  special  cases. 
Chem.  News,  XXIX  (1874),  127-128. 

3  A.  H.  Church:    A  test  of  specific  gravity.     Mineralog.  Mag.,  I  (1877),  237-238. 

4  J.  Thoulet :    Separation  des  elements  nonferrugineux  des  roches  fondee  sur  leur  difference 
de  poids  specifique.     Comptes  Rendus,  LXXXVI  (1878),  454-456. 

Idem:  Separation  mechanique  des  elements  miner alogiques  des  roches.  Bull.  Soc.  Min. 
France,  II  (1879),  17-24. 

5  V.  Goldschmidt:     Ueber  Verwendbarkeit  einer  Kaliumquecksilberjodlosung  bei  miner- 
alogischen  und  petrographischen  Unter suckling.     Neues  Jahrb.,  B.B.  I  (1881),  179-238. 


520  '  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  454 

dissolved.  So  long  as  the  proportions  of  HgI2  and  KI  remain  approximately 
as  given  above,  the  solution  may  be  diluted  with  water  in  any  amount  to  a 
minimum  density  of  i.o,  and  it  may  be  restored  by  evaporation  over  a  water- 
bath  to  its  maximum  value.  A  small  excess  of  KI,  according  to  van  Werveke1 
is  beneficial  rather  than  harmful.  If  either  salt  is  in  excess,  it  crystallizes 
out,  the  mercuric  iodide  as  a  yellow  hydrous  double  salt  in  needle-like  crystals, 
or  the  potassium  iodide  in  small  cubes.  If  the  HgI2  is  greatly  in  excess,  the 
crystallization  may  take  place  suddenly  upon  evaporation,  and  the  solution 
turn  into  a  stiff,  felty  mass  of  fine  needles.  Should  this  take  place,  a  KI 
solution,  and  not  water  alone,  should  be  added.  If  the  solution  remains  in 
contact  with  air  for  a  long  period,  both  salts  may  separate. 

In  spite  of  its  high  specific  gravity,  the  solution  may  be  filtered  readily 
through  filter  paper.  It  should  be  transparent  and  of  a  yellowish-green 
color.  After  long  use  the  solution  may  turn  reddish-brown,  due  to  the 
separation  of  iodine.  It  may  be  restored  to  its  original  condition  by  the 
addition,  with  stirring  during  evaporation,  of  a  small  quantity  of  pure 
mercury. 

The  density  of  Sonstadt's  solution  varies  with  the  humidity  of  the  air.  It 
reaches  its  maximum  of  3.196  in  winter.  In  damp  summer  weather  it  may 
be  as  low  as  3.17.  If  exposed  to  air,  the  specific  gravity  of  the  concentrated 
solution  changes  but  slightly;  when  diluted,  it  rapidly  takes  up  or  gives  off 
water  and  changes  in  value,  the  maximum  changes  taking  place  when  the 
specific  gravity  is  between  2.0  and  2.5. 

Metals  and  organic  substances  such  as  dust  or  filter  paper,  act  upon  the 
solution.  It  is  therefore  necessary  to  remove  carefully,  with  a  magnet,  all 
chips  of  iron  derived  from  mortar  or  hammer  which  may  have  become  mixed 
with  the  powder,  and  care  should  be  taken  not  to  use  metallic  forceps  in 
removing  minerals  from  the  solution.  It  has  the  further  disadvantage  of 
being  very  poisonous,  and  of  corroding  the  skin. 

To  determine  the  specific  gravity  of  a  substance  whose  density  is  between 
3.196  and  i.o,  about  25  to  40  c.c.  of  the  solution  are  placed  in  a  narrow  beaker. 
The  mineral  is  crushed  in  a  steel  mortar  and  passed  through  a  series  of  fine 
mesh  sieves.  The  coarsest  material  which  appears  homogeneous  under  the  mi- 
croscope should  be  used.  If  only  the  finest  powder  appears  to  be  uniform, 
it  should  be  separated  from  the  dust  by  washing.  The  homogeneous  material 
is  now  thrown  into  the  heavy  solution  and  water  is  added  from  a  burette, 
drop  by  drop,  with  constant  stirring,  until  the  mineral  remains  suspended 
where  placed.  If  too  much  water  has  been  added,  a  little  of  the  concentrated 
solution  will  restore  the  density.  The  addition  of  a  single  drop  of  water  has 
a  marked  effect  upon  the  concentrated  solution.  For  example,  if  one  uses  as 

1  Leopold  van  Werveke:  Ueber  Regeneration  des  Kaliumquecksilberlb'sung  und  iiber 
einen  einfachen  Apparat  zur  mechanischen  Trennung  mittelst  dieser  Losung.  Neues  Jahrb., 
1883  (II),  86-87. 


ART.  455]  DETERMINATION  OF  SPECIFIC  GRAVITY  521 

much  as  59  c.c.  of  it,  one  drop  of  water  changes  the  density  from  3.196  to 
3.194.  It  is  advisable,  therefore,  to  add  a  dilute  solution  of  the  preparation 
instead  of  pure  water  when  working  with  that  which  is  concentrated. 

To  restore  a  dilute  solution  to  its  maximum  density,  it  should  be  evaporated 
over  a  water-bath.  During  the  concentration  it  sometimes  happens  that 
crystallization  begins.  This  may  be  prevented  entirely,  according  to  Las- 
peyres,1  if  the  concentration  is  carried  on  over  the  water-bath  only  until  a 
piece  of  glass  floats.  Final  concentration,  until  a  piece  of  tourmaline  floats, 
may  be  carried  on  under  an  air  pump,  or  in  a  desiccator  in  which  some  calcium 
chloride  has  been  placed. 

Owing  to  variation  in  the  solution,  it  is  not  possible  to  determine  the 
specific  gravity  of  a  mixture  by  measuring  the  amount  of  water  which  was 
added.  The  density  should  therefore  be  determined  by  one  of  the  methods 
described  below. 

The  initial  cost  of  Sonstadt's  solution  is  considerable,  but  since  it  may  be 
used  over  and  over  and  there  is  very  little  waste,  it  does  not  amount  to  a  great 
deal  in  the  end.  At  the  present  time  the  cost  of  potassium  mercuric  iodide 
crystals  is  about  70  cents  per  ounce,  and  the  solution  costs  $1.65  per  100 
grm.,  duty  free. 

The  relation  of  the  density  of  Sonstadt's  solution  to  the  refractive  index 
has  been  given  above.2 

455.  Klein  Solution  (1881). — The  heavy  fluid  proposed  by  D.  Klein3  is  an 
aqueous  solution  of  cadmium  borotungstate  (^CdCOH^-I^Oa'gWCV^I^O). 
The  process  of  preparation  is  quite  complicated,  and  is  given  by  Edwards4 
as  follows: 

The  apparatus  necessary  are  two  large  porcelain  evaporating  dishes  10 
in.  in  diameter,  two  6  in.  in  diameter,  and  two  3  in.  in  diameter,  two  beakers 
10  in.  deep  and  two  6  in.  deep,  and  a  glass  funnel.  The  operation  should  be 
performed  under  a  hood  to  carry  off  the  fumes.  The  weights  given  below 
will  make  160  grm.  of  cadmium  borotungstate  or  about  50  c.c.  of  the  solution. 

1  H.  Laspeyres:     Vorrichtung  zur  Scheidung  von  Miner  alien  mittelst  schwerer  Losung. 
Zeitschr.  f.  Kryst.,  XXVII  (1896),  45. 

2  Art.  215. 

Besides  the  references  given  above  see  also  Rapp:  Erfahrungen  bei  der  Anwendung  der 
Thoulet'schen  Fliissigkeit.  Berichte  Versam.  oberrhein.  geol.  Vereines,  Stuttgart  XVI 
(1883), ii.* 

3  Daniel  Klein:    Sur  la  separation  mecanique  par  voie  humidq  des  miner  aux  de  densite. 
injerieure  a  3.6.    Bull.  Soc.  Min.  France,  IV  (1881),  140-155. 

Idem:  Sur  une  solution  de  densite  3.28  propre  a  r analyse  immediate  des  roches.  Comptes 
Rendus,  XCIII  (1881),  318-321. 

Review  by  H.  Rosenbusch  of  both  preceding  articles.     Neues  Jahrb.,  1882  (II),  180-191. 

4  W.  B.  D.  Edwards:    On  the  preparation  of  a  cheap  heavy  liquid  for  the  separation  of 
minerals.     Geol.  Mag.,  VIII  (1891),  273-275. 


522  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  455 

Dissolve  450  grm.  of  crystallized  sodium  tungstate  in  as  little  boiling 
water  as  possible.  When  quite  dissolved  add  675  grm.  of  boric  acid  in  small 
crystals,  a  little  at  a  time  and  with  constant  stirring.  This  should  be  done  in 
a  large  beaker.  When  entirely  dissolved  the  solution  should  be  poured  into 
a  large  evaporating  dish  and  put  aside,  covered  up  from  dust,  in  a  place  where 
it  will  not  be  disturbed  or  shaken.  In  about  twenty-four  hours  or  longer,  the 
liquid,  which  is  of  a  light  purple  color,  should  be  poured  off  quickly  from  the 
crystals  into  another  evaporating  dish.  These  crystals  should  be  in  the  form 
of  a  hard  solid  deposit  at  the  bottom  of  the  dish.  They  should  be  washed 
with  hot  water  three  or  four  times,  the  washings  being  added  to  the  mother 
liquor.  The  latter  will  now  probably  be  in  a  thick  pasty  condition  due  to 
the  formation  of  small  crystals,  which  will  be  found  to  dissolve  on  heating  the 
dish  and  its  contents  on  a  water-bath.  At  the  same  time  about  half  the  water 
can  be  driven  off,  care  being  taken  not  to  go  so  far  that  a  crust  begins  to  form 
on  the  surface  of  the  hot  liquid.  The  solution  is  again  set  aside  as  before  and 
left  to  cool  and  crystallize.  The  liquid  is  poured  off  into  one  of  the  smaller 
dishes  and  the  crystals  washed  as  before.  This  process  of  crystallization  is 
gone  through  again  until  a  piece  of  orthoclase  will  float  on  the  liquid,  the 
principal  point  being  always  to  make  the  polyborates  of  soda  crystallize  out 
either  as  single  large  crystals  or  as  a  hard  crystalline  crust.  It  is  impossible 
to  separate  and  wash  the  crystals  if  they  are  very  small.  Owing  to  the  high 
density  of  the  liquid,  in  the  later  stages  a  longer  time  is  necessary  for  the  so- 
dium borates  to  crystallize  out  than  at  first.  A  piece  of  glass  or  feldspar 
will  be  found  to  float  when  the  liquid  has  been  evaporated  down  to  about 
220  c.c. 

The  next  process  is  to  heat  the  sodium  borotungstate  on  the  water-bath 
to  1 00°  C.,  pour  into  a  large  beaker,  and  add  a  boiling  saturated  solution  of 
barium  chloride.  This  should  be  done  carefully,  a  little  at  a  time,  stirring 
the  solution  at  the  same  time.  The  barium  chloride  solution  should  consist 
of  150  grm.  of  crystals  in  about  100  c.c.  of  distilled  water.  A  dense  white 
precipitate  forms  in  pouring  this  solution  into  the  sodium  borotungstate, 
and  this  precipitate  should  be  stirred  for  some  minutes  so  as  to  thoroughly 
mix  the  two  liquids.  After  a  few  minutes,  hot  water  should  be  added  and 
the  precipitate  stirred  up  thoroughly.  In  a  short  time  the  supernatant  liquid 
can  be  siphoned  off.  This  washing  process  should  be  repeated  some  ten  or 
fifteen  times.  The  white  precipitate  is  next  transferred  to  a  large  evaporating 
dish,  and  about  300  c.c.  of  dilute  HC1  added  (i  :  10  H2O).  The  mixture  of 
precipitate  and  solution  is  now  evaporated  to  dryness  on  a  water-bath,  about 
40  c.c.  of  strong  HC1  being  added  toward  the  end.  The  dried  mass  is  then 
treated  with  about  300  c.c.  of  hot  distilled  water,  the  former  being  thoroughly 
broken  up  into  fine  powder  with  a  glass  rod  flattened  at  one  end. 

The  green  sediment  of  tungstic  hydrate  is  filtered  off  and  washed,  the  wash- 
ings being  added  to  the  solution  of  barium  borotungstate.  The  liquid  is 


ART.  455]  DETERMINATION  OF  SPECIFIC  GRAVITY  523 

evaporated  down  and  allowed  to  stand.  Yellow  crystals  are  formed,  and  with 
a  little  care  these  can  and  should  be  obtained  as  single  large  crystals.  The 
latter  crystallize  in  two  forms,  one  as  modified  tetragonal  prisms  with  well- 
developed  basal  planes,  and  the  other  in  flattened  hexagon-like  forms. 

Nearly  the  whole  of  the  barium  borotungstate  can  be  obtained,  the 
mother-liquor  being  evaporated  down  a  little  more  after  each  crop  of  crystals 
has  been  obtained.  Toward  the  end  of  the  process,  transparent  colorless 
platy  crystals  of  barium  borate  may  separate  out  as  well.  The  barium  boro- 
tungstate crystals  should  be  dissolved  in  water  and  recrystallized  once 
more.  They  should  then  be  dissolved  in  200  c.c.  of  distilled  water,  and  a 
solution  of  CdSC>4  added  from  a  burette  or  a  pipette,  care  being  taken  to  add 
it  very  slowly,  drop  by  drop,  as  long  as  a  precipitate  falls.  The  precipitate  of 
BaSO4  is  then  filtered  off,  and  the  filtrate  is  evaporated  in  a  porcelain  dish 
in  a  water-bath  till  a  piece  of  olivine  floats  on  the  surface. 

This  liquid  will  be  found  to  have  a  specific  gravity  of  3.46  at  60°  F.,  and 
it  takes  some  hours  before  some  of  the  salt  crystallizes  out  and  the  specific 
gravity  falls  to  3.28.  It  is  of  a  clear  yellow  color. 

The  quantity  of  cadmium  borotungstate  obtained  by  the  above  process  is 
about  1 60  grm.  or  enough  to  make  50  c.c.  of  the  solution.  According  to 
Edwards  the  cost  (in  1891,  with  sodium  tungstate  at  i  s.  for  450  grm.)  was 
2  s.  4d.  for  the  materials  and  no  account  taken  of  the  time  spent  in  preparation. 
Cadmium  borotungstate  crystals  are  listed  at  the  present  time  at  $1.50  per 
ounce,  or  the  solution  at  $8.00  a  pound. 

Klein's  solution  may  be  diluted  to  any  amount  with  water,  and  again  con- 
centrated to  its  former  density  by  evaporation  on  the  water-bath.  If  the 
final  crystals  produced  in  the  process  of  preparation  are  heated  in  a  tube 
on  a  water-bath  to  75°  C.  they  will  melt  in  their  own  water  of  crystallization, 
and  a  rather  oily  fluid  with  a  specific  gravity  of  3.55  will  be  obtained.  It  is 
not,  however,  suitable  for  the  separation  of  powders.  The  solution  with  a 
specific  gravity  of  3.36,  such  as  is  obtained  by  the  concentration  of  a  dilute 
solution  by  evaporation  until  a  crystalline  film  forms  over  the  surface,  is 
not  yet  of  oily  consistency. 

The  solution  is  not  very  poisonous,  nor  does  it  act  upon  the  skin  nor  upon 
filter  paper,  through  which  it  readily  passes.  After  repeated  use,  the  solution 
becomes  dark,  but  it  may  be  cleared,  according  to  van  Werveke,  by  the  addi- 
tion of  a  few  drops  of  peroxide  of  hydrogen.  It  possesses  the  disadvantages 
of  being  decomposed  by  metallic  lead,  zinc,  and  iron,  and  of  being  acted  upon 
by  carbonates,  wherefore  it  is  necessary,  beforehand,  to  treat  the  mineral  to 
be  examined,  with  dilute  acetic  acid. 

Mann1  found  the  keeping  qualities  of  this  solution  to  be  superior  to  Son- 
stadt's  or  Rohrbach's;  a  preparation  in  continual  use  for  a  number  of  years, 

1  Paul  Mann:  Untersuchungen  Uber  die  chemische  Zusammenselzung  einiger  Augite  aus 
Phonolithen  und  verwandten  Gesteine.  Neues  Jahrb.,  1884  (II),  179-180. 


524  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  456 

and  repeatedly  diluted  and  condensed,  showed  not  the  slightest  alteration 
from  its  original  condition. 

456.  Rohrbach  Solution,  (1879,  1883). — According  to  Karpinskij,1 
a  solution  of  barium-mercuric  iodide  was  used  for  the  determination  of 
specific  gravities  by  Suschin  in  1879,  and  described  by  Karpinskij  in  i88o.2 
The  publication,  however,  being  in  the  Russian  language,  was  seen  by  few 
investigators  and  it  was  not  until  1883,  when  Rohrbach3  rediscovered  it, 
that  the  solution  came  into  general  use. 

In  preparing  this  solution  it  is  necessary,  on  account  of  the  hygroscopic 
character  of  the  barium  iodide  and  its  rapid  decomposition  in  solution,  to 
work  quickly  until  the  double  salt  is  formed.  It  is  prepared  as  follows: 
100  parts  of  barium  iodide  and  130  parts  of  mercuric  iodide  are  rapidly  weighed 
out  and  are  shaken  up  together  in  a  dry  flask,  after  which  20  parts  of  distilled 
water  are  added  and  the  whole  is  placed  upon  an  oil-bath  previously  heated 
to  i5o°-2oo°  C.  The  salts  are  more  rapidly  dissolved  and  the  formation  of 
the  double  salt  promoted  if  the  material  in  the  flask  is  stirred  by  rapidly 
twirling  in  it  a  crutch-shaped  glass  rod,  held  between  the  fingers.  When  all 
is  dissolved,  the  solution  is  boiled  a  short  time  longer,  after  which  it  is  poured 
into  a  porcelain  evaporating  dish  and  is  placed  over  a  water-bath  until  an 
Untersulzbachthal  epidote  crystal  (G  =  3.4)  will  just  float.  A  small  amount  of 
a  yellow  double  salt  will  separate  out  on  cooling.  In  spite  of  this,  however, 
on  account  of  the  contraction  of  the  liquid,  its  specific  gravity  rises,  so  that, 
when  cold,  a  topaz  crystal  (6  =  3.55)  w^l  fl°at  upon  it.  Since  the  solution 
acts  upon  filter  paper  and  converts  it  into  a  parchment-like  substance,  it  is 
not  possible  to  filter  off  the  clear  liquid.  It  should  be  left  undisturbed  for  a 
few  days  in  a  closed  flask  and  then  decanted.  The  solution  is  of  a  clear  yellow 
color  but  it  becomes  darker  upon  heating.  It  boils  at  145°  C.  and  gives  off 
steam  and  red  mercuric  iodide  vapor.  It  has  a  high  refractive  index. 4 

Rohrbach's  solution  is  not  affected  by  carbonates,  but  it  is  hygroscopic  and 
should  be  kept  in  closed  vessels.  It  is  also  extremely  poisonous.  Its  great 
disadvantage,  however,  is  the  difficulty  of  diluting  it,  for  on  mixing  with 
water  at  ordinary  temperatures,  crystals  of  red  mercuric  iodide  separate,  and 
these  will  not  dissolve  again  in  the  cold  solution.  The  reduction  in  density 
must  therefore  be  made  by  adding,  very  slowly,  a  dilute  solution  of  the 
same  preparation,  the  latter  being  made  by  adding  water,  drop  by  drop,  with 

1  Review  by  Ernst  Kalkowsky  of  A.  Karpinskij.     Cit.  supra. 

2  Trudy  St.  Petersburgh  Obschtsch.  jestjestw.,  XI  (1880),  146.* 

3  Carl  Rohrbach:     Ueber  eine  neue  Flussigkeit  ion  hohem  specifischen  Gewicht,  hohem 
Brechungsexponenten  und  grosser  Dispersion.     Wiedem.  Ann.,  N.  F.  XX  (1883),  169-174. 

Idem:  Ueber  die  V  erwendbarkeit  einer  Baryumquecksilberjodid-Losiing  zu  petrographi- 
schen  Zwecken.  Neues  Jahrb.,  1883  (II),  186-188. 

4  Art.  217. 


ART.  457]  DETERMINATION  OF  SPECIFIC  GRAVITY  525 

constant  stirring,  to  a  portion  of  the  solution  heated  nearly  to  the  boiling- 
point,  or  by  adding  carefully  a  thin  stratum  of  water  to  a  portion  and  leav- 
ing it,  for  twenty-four  hours,  to  mix  by  diffusion. 

The  minerals  whose  specific  gravities  are  to  be  determined  must  be  abso- 
lutely dry.  Upon  removing  them  from  the  solution  they  must  be  washed, 
not  with  pure  water,  but  with  water  to  which  a  few  drops  of  potassium  iodide 
have  been  added. 

Instead  of  making  the  complete  separation  of  rock  constituents  by  means 
of  the  Rohrbach  solution,  it  is  advisable  to  separate  first  the  heavier  from  the 
lighter  constituents  by  means  of  the  Sonstadt,  and  use  the  Rohrbach  only 
for  those  whose  density  is  greater  than  2.9.  Since  the  solution  is  hygro- 
scopic, the  separation  should  be  performed  in  closed  vessels,  such  as  the 
Thoulet  or  Harada  tubes. 

The  cost  of  the  components  of  this  solution,  at  the  present  time,  is  about 
35  cents  an  ounce. 

457.  Methylene  Iodide  (Brauns)  (1886). — So  long  ago  as  1873,  Sonstadt1 
used  ethyl  iodide  (C2H5I,  with  0  =  1.93),  prepared  from  commercial  methy- 
lated spirits,  therefore  containing  a  few  per  cent,  of  methyl  iodide,  and  having 
a  density  of  about  2.0.  For  the  determination  of  density  he  diluted  it  with 
bisulphide  of  carbon  or  chloroform,  preferably  the  former  since  it  is  less  vola- 
tile. To  prevent  the  liquid  from  becoming  discolored  by  the  separation  of 
iodine,  he  placed  in  it  magnesium  (or  copper)  wire  or  filings. 

The  first  use  made  of  methylene  iodide  (CH^)  was  by  Brauns,2  in  1886. 
This  substance  is  a  thin,  light  yellow  fluid  of  high  refractive  index,  boiling  at 
1 80°  C.  with  partial  decomposition,  and  freezing  at  5°  C.  It  can  be  diluted 
with  neither  water  nor  alcohol,  but  may  be,  in  all  proportions,  with  benzol. 
It  is  unaltered  by  exposure  to  the  atmosphere  and  is  slow  to  evaporate  when 
concentrated,  so  that  one  can  work  for  hours  with  no  apparent  change  in  the 
refractive  index  or  specific  gravity.  When  diluted  it  changes  its  refractive 
index  and  density  rapidly  by  the  evaporation  of  the  benzol.  It  does  not  act 
upon  the  skin,  metals,  nor  carbonates,  but  is  decomposed  by  sulphur.  The 
cost  is  rather  more  than  the  other  heavy  fluids  already  mentioned,  being, 
at  the  present  time,  about  $3.25  per  100  grm.,  duty  free,  as  against  $1.65 
for  the  same  amount  of  Sonstadt's. 

Both  refractive  index  and  specific  gravity  change  rapidly  with  change  in 
temperature,  as  may  be  seen  from  the  table  in  Article  218,  and  the 
following : 

1  Op.  cit. 

2  R.  Brauns:     Ueber   die    Verwendbarkeit  des  Methylenjodids  bei  petrographischen  und 
optischen  Uniersuchungen.     Neues  Jahrb.,  1886  (II),  72-78. 

C.  Chelius:  Zur  Benutzung  des  Methylenjodids.  Notizbl.  Ver.  f.  Erdk.  Darmstadt, 
1800  (4),  16.* 


526 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  458 


TABLE  SHOWING  THE  RELATION  BETWEEN  TEMPERATURE  AND  SPECIFIC  GRAVITY 

OF  METHYLENE  IODIDE 


Temp. 

c 

i 

Temp. 

G 

Temp. 

G 

Temp. 

G 

5°C. 

3.3485 

11°  C. 

3-3353 

17°  C. 

3.3221 

23°  C. 

3-3089 

6 

3-3463 

12 

3-3331 

18 

3-3I99 

24 

3.3067 

7 

3-3441 

13 

3-3309 

19 

3.3177 

25 

3-3045 

8 

3-34I9 

14 

3-3287 

20 

3-3I55 

;  

9 

3-3397 

15 

3-3265 

21 

3-3I33 

33 

3.2890 

10 

3-3375 

16 

3-3243 

22 

3-3IH 

74 

3.1890 

The  chief  advantage  of  this  fluid,  especially  for  mechanical  separation  of 
minerals,  is  its  mobility,  whereby  even  fine  powders  may  be  separated,  a  thing 
impossible  with  Sonstadt's  or  Klein's.  The  cleaning  of  the  recovered  powder 
is  also  very  simple,  a  washing  in  benzol  being  usually  all  that  is  necessary. 
If  a  little  of  the  methylene  iodide  should  remain,  it  may  be  driven  off  by 
gentle  heat.  It  possesses  the  further  advantage  of  being  usable  for  the  sepa- 
ration of  minerals  soluble  in  water. 

To  concentrate  a  dilute  solution  it  is  only  necessary  to  place  it  on  a  water- 
bath,  or  to  expose  it  in  shallow  vessels  to  an  air  current,  which  will  rapidly 
volatilize  the  benzol.  Some  of  the  methylene  iodide  will  be  lost  at  the  same 
time,  a  disadvantage  on  account  of  the  expense.  Upon  exposure  to  sunlight 
or  heat  the  fluid  may  turn  brown  by  the  separation  of  iodine.  It  may  be 
cleared  by  shaking  it  up  with  dilute  potassium  hydroxide  (KOH),  washing 
with  clean  water,  drying  by  the  addition  of  a  piece  of  calcium  chloride,  and 
filtering.  It  has  no  effect  upon  filter  paper  and  readily  passes  through.  A 
simpler  method  of  clearing1  is  to  reduce  the  temperature  to  5°  C.,  whereupon 
it  solidifies,  leaving  a  small  amount  of  a  brown  liquid,  which  may  be  poured 
off  and  cleared  when  convenient  with  potassium  hydroxide.  The  amount  nee- 
necessary  to  so  clear,  however,  will  not  be  great.  The  crystallized  portion, 
upon  melting,  will  be  perfectly  clear  and  transparent.  Another  method  of 
clearing  is  given  by  Schroeder  van  der  Kolk,2  who  says  that  the  iodine  may  be 
removed  with  copper. 

458.  Retgers'  Heavy  Fluids  (1889). — A  great  number  of  experiments  were 
made  by  Retgers3  to  obtain  fluids  having  higher  densities  than  any  previously 
used.  He  found  that  after  concentrating  Sonstadt's  solution  on  the  water- 
bath  until  a  surface  film  was  produced,  he  could,  by  stirring,  dissolve  flakes 
of  iodine  in  it,  and  thus  obtain  a  black,  opaque  liquid.  Upon  cooling,  a  cer- 

1  R.  Brauns:  Eine  einfache  Methode  Methylenjodid  zu  kldren.  Neues  Jahrb.,  1888  (I), 
213-214. 

*  J.  L.  C.  Schroeder  van  der  Kolk:  Tabellen  zur  mikroskopischen  Bestimmung  der  Min- 
er alien.  Wiesbaden,  1900,  13. 

8  J.  W.  Retgers:  Ueber  schwere  Flussigkeiten  zur  Trennung  von  Miner  alien.  Neues 
Jahrb.,  1889  (II),  185-192. 


ART.  458]  DETERMINATION  OF  SPECIFIC  GRAVITY  527 

tain  part  of  the  iodine  separated,  but  the  decanted  liquid  itself  had  a  specific 
gravity  of  3.30-3.40. 

Evaporating  Rohrbach's  solution  on  the  water-bath  to  the  formation  of 
a  crystalline  surface  film  and  saturating  with  iodine  gives  a  nearly  opaque 
fluid  from  which,  on  cooling,  a  portion  crystallizes  out.  The  decanted  liquid 
has  a  density  of  3.6  to  3.65  at  20°  C.  In  one  experiment  a  value  of  3.70  was 
obtained. 

Iodine  and  sulphur  had  been  dissolved  in  methylene  iodide  so  long  ago  as 
1888  by  Bertrand1  to  obtain  a  fluid  of  high  refractive  index.  Retgers  used 
iodine  alone  and  obtained  an  opaque,  black  fluid,  more  mobile  than  the  two 
solutions  just  mentioned,  and  of  a  refractive  index  of  3.543-3.549.  It  does 
not  alter  in  air. 

All  of  the  above,  however,  are  of  practically  no  use  in  the  determination  of 
specific  gravities  since  they  are  opaque,  and  the  point  of  suspension  of  the 
mineral  fragments  cannot  be  seen.  To  a  certain  extent  they  may  be  used  as 
separating  fluids.2  None  can  be  filtered  through  paper. 

A  transparent  fluid  of  high  specific  gravity  was  prepared  by  Retgers3 
by  slightly  warming  methylene  iodide  (CH2l2)  and  dissolving  in  it  as  much 
iodoform  (CHI3)  as  it  would  take  up.  Although  much  was  dissolved,  a  con- 
siderable amount  recrystallized  upon  cooling.  Generally  some  decomposition 
of  the  iodoform  takes  place  and  gives  the  solution  a  dirty  brown  color.  It 
may  be  cleared,  however,  by  shaking  with  potassium  hydroxide,  leaving  a 
deep  yellow,  transparent  fluid  with  a  density  of  3.456  at  24°  C.  The  solution 
thus  prepared  will  still  dissolve  iodine,  and  when  saturated  and  cold  has  a 
density  of  3.60-3.65.  It  is  opaque. 

Among  other  solutions  prepared  by  Retgers4  are  a  saturated  solution  of 
SnI4  in  AsBr3,  giving  a  density  of  3.73  at  i5°C.,  and  a  saturated  solution 
of  Asls  and  SbI3  in  a  mixture  of  AsBr3  and  CH2I2,  giving  a  density  of  3.70 
at  20°.  A  solution  of  selenium  in  selenium  bromide  (SeBr)  would  probably 
have  a  density  of  3.70.  Lead  tetra-chloride  (PbCl4)  is  a  clear  yellow 
fluid  which  solidifies  at  —15°  C.  and  has  a  density  of  3.18  at  o°.  The 
analogous  PbBr4  probably  has  a  density  of  3.5  and  is  also  transparent. 

As  a  result  of  his  experiments,  Retgers  came  to  the  conclusion  that  it  is 
probably  hopeless  to  expect  to  find  a  fluid  having  a  density  greater  than  4.0, 
mercury  being  an  exception. 

1  Emile  Bertrand:    Liquides  d'indices  superieurs  a  1.8.     Bull.  Soc.  Min.  France,  XI 
(1888),  31. 

2  See  Chapter  XXXIX,  infra. 

3  Op.  cit. 

4  J.   W.    Retgers:    Die  Darstellung  neuer  schwerer  Flussigkeiten.     Zeitschr.  f.  physik. 
Chemie.  XI  (1893),  328-344. 

Idem:  Ueber  die  miner alogische  und  chemische  Zusammensetzting  der  Dunensande  Hol- 
lands und  tiber  die  Wichtigkeit  von  Fluss-  und  Meeressanduntersuchungen  im  Allgemeinen. 
Neues  Jahrb.,  1895  (I),  16-74,  especially  28-31. 


528 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  459 


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ART.  462] 


DETERMINATION  OF  SPECIFIC  GRAVITY 


529 


460.  Schroeder  vanderKolk  (1895). — Schroeder  van  der  Kolk,1  in  1895, 
used  bromoform  (CHBr3)   as  a  heavy  solution  for  the  separation  of  the 
constituent  minerals  of  sands.     It  has  a  density  of  2.88  and  is  useful  for  the 
separation  of  many  minerals.     It  is  much  less  expensive  than  methylene 
iodide,  costing  about  10  cents  an  ounce.    Its  melting-point  is  2.5°  C.,  its 
boiling-point  151°,  and  its  refractive  index  by  sodium  light  at  15°,  1.588. 
It  is  not  decomposed  by  air  or  light.    It  should  be  used  without  dilution. 

461.  Muthmann  (1898). — Muthmann2  used  symmetrical  acetylene  tetra- 
bromide  (CHBr2— Br2HC),  with  a  specific  gravity  of  3.01,  as  a  heavy  solution. 
Its  melting-point  is  below  —  20°  C.  and  its  boiling-point  137°  under  a  pressure 

10°      20'      30°      40°      50°      60°       70°      80°      90°    100°C 


D.UU 

4.50 

4.00 
8.50 

3.00 
2  Jin 

Densities  of  saturat 
ts'olutidn 

ed 

|of   | 
Ilium  formiat* 

, 

/ 

-^ 

atd 

iffere 
aft< 

nt  tempera 
r  Clerici. 

;ures 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

FIG.  719. — Diagram  showing  densities  of  a  saturated  solution  of  thallium  formiate  at  different  tempera- 
tures.    (After  Clerici.) 

of  3.6  of  mercury.  It  may  be  diluted  with  ether,  benzol,  or  toluol  in  any 
proportions,  and  may  be  restored  to  its  original  density  by  evaporation.  It  is 
decomposed  by  neither  air  nor  minerals,  not  even  by  ores.  Since  its  density 
is  almost  exactly  3.0,  it  is  useful  as  one  of  a  series  of  density  fluids.  Minerals 
removed  from  this  fluid  may  be  cleaned  by  washing  with  ether.  It  costs  less 
than  10  cents  an  ounce. 

462.  Clerici  (1907). — -Clerici3  examined  numerous  thallium  salts  of  the 

1  J.  L.  C.  Schroeder  van  der  Kolk:    Beilrag  zur  Kartirung  der  quartaren  Sande.     Neues 
Jahrb.,  1895  (I),  272-276,  especially  274. 

2  W.   Muthmann:    Ueber  eine  zur   Trennung  von  Miner algemischen  geeignete  schwere 
Fliissigkeit.     Zeitschr.  f.  Kryst.,  XXX  (1898),  73-74. 

3  Enrico   Clerici:    Preparazione   di  liquidi   per   la   separazione   del   miner  all.     Riend. 
Acad.  Lincei.  Roma  (Ser.  5),  XVI  (1907),  i  Semsetre,  187-195. 

34 


530  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  463 

organic  acids  and  found  the  one  best  adapted  as  a  heavy  solution  to  be 
thallium  formiate,  aqueous  solutions  of  which  are  colorless  and  free-flowing 
as  water.  The  amount  of  the  salt  necessary  for  saturation  depends  upon  the 
temperature,  5.0  grin,  being  soluble  in  i  c.c.  of  water  at  10°  C.  The  density 
of  the  saturated  solution  varies  with  the  amount  of  salt  dissolved  (Fig.  719), 
being  2.58  at  — 10°  C.,  2.86  at  o°,  3.14  at  10°,  3.40  at  20°,  3.64  at  30°,  3.87  at 
40°,  4.11,  at  50°,  4.32  at  60°,  4.50  at  70°,  4.67  at  80°,  and  4.76  at  90°.  At  95° 
thallium  formiate  melts  and  has  a  density  great  enough  to  float  ilmenite. 
The  solution  is  miscible  with  water. 

To  regain  the  thallium,  the  filtered  liquid  is  concentrated  on  a  water-bath, 
or  H2SO4  is  added  and  the  thallium  precipitated  on  a  zinc  plate. 

463.  Joly  (1886). — Joly1  determined  the  specific  gravity  of  mineral  frag- 
ments by  placing  them  on  a  piece  of  previously  weighed  paraffine,  care- 
fully melting  them  in,  and  proceeding  as  by  the  method  of  Thoulet. 

464.  Hubbard  (1887). — A   method  for  approximately   determining   the 
specific  gravity  of  a  mineral  was  given  by  Hubbard.2    He  simultaneously 
placed  the  unknown  mineral  fragments  and  a  number  of  other  minerals  of 
known  density  in  a  cadmium  borotungstate  solution  (£  =  3.3).     By  noting 
the  rapidity  with  which  the  different  minerals  sank,  he  determined  their  rela- 
tive densities. 

465.  Streng  (1887). — To  determine  the  specific  gravity  of  minerals  heavier 
than  the  immersion  fluid,  Streng3  made  use  of  small  glass  cups  constructed 
from  glass  tubing.     They  were  made  as  thin  as  possible,  from  5  to  6  mm.  deep 
and  5  mm.  in  diameter  at  the  top.     To  the  bottom  of  each,  three  platinum 
wires  were  fused  to  serve  as  counterpoises  to  keep  the  vessel  erect.     Into 
the  cup  a  fragment  of  the  mineral  to  be  tested  was  placed,  and  the  cup  was 
filled  with  the  heavy  solution,  all  bubbles  being  removed  by  means  of  a  piece 
of  platinum  wire.     The  cup  and  mineral  were  now  placed  in  the  heavy  solu- 
tion which  was  diluted,  with  constant  stirring,  until  they  were  suspended. 


1 J.  Joly:  On  a  method  of  determining  the  specific  gravity  of  small  quantities  of  dense  or 
porous  bodies.  Phil.  Mag.  XXVI  (1888),  29-33. 

Idem:  On  a  method  of  determining  the  specific  graiity  of  small  quantities  of  dense  or  porous 
bodies.  Proc.  Roy.  Dublin  Soc.,  V  (1886),  41-47. 

2L.  L.  Hubbard:  Beitrdge  zur  Kenntnis  der  Noseanfiihrenden  Auswiirflinge  des  Laacher 
Sees.  T.  M.  P.  M.,  VIII  (1887),  390. 

3  A.  Streng:  Ueber  die  Bestimmung  des  specifischen  Gewichts  schwerer  Miner  alien. 
Ber.  oberhess.  Gesell.  Giessen,  XXV  (1887),  110-113. 


ART.  467]  DETERMINATION  OF  SPECIFIC  GRAVITY  531 

If  m  =  absolute  weight  of  the  cup, 

5  =  sp.  gr.  of  the  cup, 
m'  =  absolute  weight  of  the  mineral, 
5  =  sp.  gr.  of  mineral  and  float,  therefore  also  of  the  heavy  solution  when  they  are 

suspended, 

x  =  desired  sp.  gr.  of  the  mineral, 
we  have, 

m     m      m~\~m  S 


s        x 


m  f      S\ 

—  — /(i 

m  x      s  ] 


466.  Retgers  ( 1889). — Retgers1  pointed  out  that  in  Streng's  method  errors 
were  likely  to  occur  through  the  incomplete  mixing  of  the  heavy  solution 
during  dilution,  and  through  the  probability  that  the  solution  within  the  cup 
remains  heavier  than  that  in  the  surrounding  beaker.     The  absolute  weight 
of  the  cup  is  also  too  great,  as  comparedVith  the  mineral,  to  give  very  accurate 
results,  except  with  very  large  fragments.     To  remedy  this,  Retgers  made 
glass  clips  by  bending  fine  glass  threads  into  the  form  of  horseshoes,  the  spring 
of  the  glass  keeping  the  ends  together  or  pressed  against  the  mineral  fragment 
whose  specific  gravity  is  to  be  determined.     A  great  number  of  these  clips 
of  various  sizes  and  forms  may  quickly  be  constructed,  their  absolute  weight 
and  specific  gravity  being  determined  once  for  all. 

In  determining  the  density  of  a  mineral  by  the  use  of  these  glass 
clips,  it  is  essential  that  they  be  chosen  no  larger  than  necessary  to  give 
the  desired  buoyancy.  The  formula  to  be  used  is  the  same  as  in  Streng's 
method. 

467.  Behr  (1903). — Owing  to  the  fragility  of  Retgers'  glass  floats,  Behr2 
replaced  them  by  glass  tubes,  such  as  are  used  in  organic  chemistry  for  the 
determination  of  melting-points.     These  tubes  were  shortened  to  10  or  12 
mm.,  and  into  one  was  inserted  the  crystal  whose 'density  was  to  be  measured, 
its  absolute  weight  having  been  determined  previously.     To  prevent  the 
crystal  from  slipping  out,  each  end  of  the  tube  was  slightly  constricted  by 
heating  it.     The  specific  gravity  of  the  combination  was  now  determined, 
and  the  true  density  computed  by  the  formula  given  under  Thoulet's  method 
above.3 

For  the  best  results,  the  absolute  weight  of  the  tube  in  proportion  to 
that  of  the  crystal  must  not  be  too  large;  the  best  proportions  being  such  that 
the  combination  has  a  specific  gravity  but  little  less  than  the  original  immer- 
sion fluid.  During  the  process  of  dilution,  the  cylinder  must  be  taken  out 
repeatedly  so  that  the  concentration  of  the  solution  inside  becomes  the  same 
as  that  outside. 

1  J.  W.  Retgers:    Die  Bestimmung  des  specifischen  Gewichtes  in  Wasser  loslichen  Salzen- 
Zeitschr.  f.  phys.  Chemie,  IV  (1889),  180-196. 

2  J.  Behr:    Beitrdge  zn  den  Beziehungen  zivischen  eutropischen  und  isomorphen  Substanzen. 
Neues  Jahrb.,  1903  (I),  136-159,  especially  143-146. 

3  Art.  464. 


532  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  468 

DETERMINATION  or  THE  SPECIFIC  GRAVITY  or  THE  HEAVY  SOLUTION 

468.  Goldschmidt's  Method  (1881). — The  method  of  Goldschmidt1  for 
the  determination  of  the  specific  gravity  of  the  heavy  solution,  after  dilution 
to  the  point  of  suspension  of  the  mineral,  is  as  follows:  First  enough  of 
the  solution  is  poured  into  a  glass  flask  to  reach  a  mark  indicating  exactly 
250  c.c.  The  exact  point  may  best  be  determined  by  filling  the  flask 
slightly  above  the  mark,  which  should  extend  entirely  around  the  neck, 
bringing  it  to  the  level  of  the  eye  so  that  the  scratch  appears  as  a  straight 
line,  and  removing  enough  of  the  fluid,  with  a  pipette  or  piece  of  filter  paper, 
to  bring  the  lower  curve  of  the  meniscus  tangent  to  this  mark.  The  flask 
and  contents  are  now  weighed  together,  after  which  the  fluid  is  returned  to 
the  beaker  containing  the  mineral,  and  notice  is  taken  that  the  mineral  is 
again  suspended.  It  is  again  poured  to  the  25  c.c.  mark,  and  again  flask 
and  fluid  are  weighed.  The  same  process  is  repeated  a  third  time.  These 
results  are  now  averaged.  Should  the  values  differ  widely,  the  process  is 
repeated  a  fourth  time  to  see  if  an  error  has  been  made.  Note  particularly 
that  the  mineral  is  suspended  between  each  reading,  and  that  the  flask  is 
filled  exactly  to  the  mark  each  time.  Should  the  mineral  and  fluid  not  be  of 
exactly  the  same  specific  gravity  at  any  reading,  owing  to  evaporation  or 
other  cause,  the  density  is  corrected  by  adding  a  few  drops  of  water  or  of 
concentrated  solution. 

The  weight  of  the  flask  alone,  which  was  determined  once  for  all,  is  now 
subtracted  from  the  average  reading,  and  the  result  is  divided  by  25,  thus 
giving  the  weight  of  i  c.c.  of  solution.  Since  i  c.c.  of  water  weighs  i  grm., 
the  result  expresses  the  specific  gravity  of  the  solution. 

Goldschmidt  gives  several  examples,  among  which  is  the  following  for 
Herkimer  County,  New  York,  quartz. 

Flask  and  25  c.c.  fluid 77-991 

77.972 

77-9QI 

233.864 

Average 77-955 

Weight  of  flask 11.682 


25)66.273 
2.651=6 

The  triple  readings,  repeated  three  times,  gave  averages  of  77. 959,  77.955, 
and  77.981,  but  the  value  for  the  density  in  each  case  was  2.651. 

469.  Sprengel  Tube. — The  method  with  the  Sprengel  tube  (Fig.  720)  is 
very  similar.     The  heavy  liquid  is  sucked  up  into  the  tube  to  a  mark  indicating 

1 V.     Goldschmidt:     Ueber     Verwendbarkeit     einer     Kaliumquecksilberjodidldsung     bel 
miner  do  gischen  und  petrographischen  Untersnchung.     Neues  Jahrb.,BB.  I  (1881)  197-199. 


ART.  471] 


DETERMINATION  OF  SPECIFIC  GRAVITY 


533 


a  known  volume,  the  apparatus  is  hung  from  a  balance  by  means  of  a 
platinum  wire,  and  the  tube  and  contained  liquid  are  weighed  (w).  The  weight 
of  the  tube  alone  (w'),  and  of  the  tube  filled  to  the  mark  with  distilled  water 
(w"),  are  known,  therefore 

w  —  wf 


470.  Sollas'  Modification  of  the  Sprengel  tube  (1886). — -ScHas1  proposed 
a  modification  of  the  Sprehgel  tube  by  connecting  the  two  enlarged  parts  by  a 
thin  calibrated  tube.     The  method  otherwise  is  the  same. 

471.  Westphal  Balance  (1883). — One 
of  the  simplest  methods  for  the  determi- 
nation of  the  specific  gravity  of  a  liquid  is 
by  means  of  a  balance  originally  manufac- 
tured by  G.  Westphal  in  Celle,  and  now 
generally  known  as  the  Westphal  balance 
(Fig.    721).     It   was    first    described    by 
Cohen.2    The  method  of  determining  the 
specific  gravity  is  as  follows : 


PIG.  720. — Sprengel  tube.     (Central  Scientific  Co.)          FIG.     721.- 


-  Westphal     balance. 
Scientific  Co.) 


(Central 


The  fluid  whose  density  is  to  be  determined  is  placed  in  a  cylindrical 
vessel,  and  in  it  is  placed  a  weighted  sinker,  generally  made  to  serve  as  a 
thermometer  for  determining  the  temperature  of  the  fluid  as  well.  This 
sinker  is  attached  to  a  balance  so  arranged,  when  it  is  in  equilibrium,  that 
the  pointer,  shown  at  the  left,  indicates  zero.  With  any  fluid  heavier  than 
water  in  the  glass  vessel,  the  sinker  rises,  and  to  reestablish  equilibrium  a 
certain  number  of  riders  must  be  hung  upon  the  right  'arm  of  the  balance. 
This  arm  is  divided  into  ten  parts.  The  density  of  the  liquid  is  equal  to  the 
number  of  unit  riders  hung  at  the  end  of  the  arm,  plus  the  number  of  tenths 
indicated  by  a  unit  rider  hung  at  one  of  the  ten  divisions,  plus  the  number 

l\\.  J.  Sollas:  On  a  separating  apparatus  for  use  with  heaiy  fluids.  Proc.  Ro>.  Dublin 
Soc.,  N.  S.,  V  (1886-7),  621-622. 

2  E.  Cohen:  Ueber  eine  einfache  Methode  das  specifische  Geivicht  einer  Kaliumquecksilber- 
iodidlosttng  zu  bestimmen.  Neues  Jahrb.,  1883  (II),  87-89. 


534  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  472 

of  one-hundredths  indicated  by  a  one-tenth  unit  rider,  plus  the  number  of 
one-thousandths  indicated  by  a  one-hundredth  rider.  If  the  volume  of 
the  sinker  is  exactly  i  c.c.,  the  unit  rider  will  be  of  i  grm.  weight,  the  second 
of  o.i  grm.,  and  the  third  of  o.oi  grm. 

With  this  balance,  determinations  may  be  made  very  quickly,  and  the 
results  are  accurate  to  2  in  the  third  decimal  place.  The  chief  source  of  error  is 
the  adhesion  of  air  bubbles  to  the  sinker.  They  should  be  carefully  removed 
by  means  of  a  glass  rod  or  platinum  wire.  All  of  one  set  of  determinations 
should  be  made  at  the  same  temperature. 

472.  Salomon's  Apparatus  (1891). — -An  apparatus,  based  on  the  principle 
that  two  fluids  of  different  specific  gravities  placed  in  communicating  tubes 
will  stand  at  heights  inversely  proportional  to  their  densities,  was  designed 
by  Salomon.1    The  operation  of  determining  the  density  of  a  fluid  by  this 
apparatus,  however,  is  much  more  complicated  than  by  the  Westphal  bal- 
ance, and  the  results  are  less  accurate. 

473.  Sollas  Hydrostatic  Float  (1891). — Another  method  of  determining 
the  specific  gravity  of  a  fluid  is  by  means  of  a  hydrostatic  float,  such  as 
that  proposed  by  Sollas.2    This  consists  of  a  thin  glass  tube,  drawn  out  at 
one  end  into  a  capillary  tube,  and  closed  at  the  other.     The  closed  end 
contains  enough  mercury  to  cause  the  instrument  to  float  in  a  vertical  posi- 
tion, the  length  of  the  capillary  tube  projecting  from  the  immersing  fluid 
serving  as  a  measure  of  the  specific  gravity  of  the  latter.    The  instrument 
may  be  calibrated  by  placing  it  in  several  fluids  of  known  densities,  marking 
the   projection,    and    interpolating    values.     By    employing    several    such 
hydrostatic  floats  with  different  ranges  in  values,  results  accurate  to  the 
third  decimal  place  may  be  obtained  rapidly. 

474.  Merwin's  Method  by  Refractive  Indices  (1911). — A  method  for 
determining  the  density  of  Rohrbach's  solution  by  means  of  its  refractive 
index — 'the  solution  having  a  fixed  density  for  a  given  refractive  index — • 
was  given  by  Merwin.3    The  advantage  of  this  method  is  that  the  amount 
of  liquid  required  is  not  so  great  as  in  some  of  the  other  methods.    It  is 
necessary,  however,  to  have  a  refractometer  to  measure  the  indices.    The 
purity  of  the  solution  may  be  checked  by  bringing  it  to  the  density  of  clear 
quartz  (£=2.6495)  and  determining  its  index  of  refraction  for  sodium  light 
at  20°  C.     It  should  be  1.6208.     The  values  found  are  given  in  the  following 
table  and  are  graphically  represented  in  Fig.  722. 

1  W.  Salomon:    Ein  neuer  Apparat  zur  Bestimmung  des  specifischen  Gewichtes  von  Fliissig- 
keiten.     Neues  Jahrb.,  1891  (II),  214-220. 

2  W.  J.  Sollas:  Contributions  to  a  knowledge  of  the  granites  ofLeinster.   Read  Nov.  30,  1889. 
Trans.  Roy.  Irish  Acad.,  Dublin,  XXIX  (1887-1892),  427-514,  especially  430-431. 

3  H.  E.  Merwin:    A  method  of  determining  the  density  of  minerals  by  means  of  Rohrbach's 
solution  hating  a  standard  refractive  index.     Amer.  Jour.  Sci.,  XXXII  (1911),  425-432. 


ART.  470] 


DETERMINATION  OF  SPECIFIC  GRAVITY 


535 


Density  at  20°  C. 

Refractive  index 

3-449 

.7686 

3-396 

-7590 

3.246 

.7312 

3.180 

.7195 

3.046 

.6944 

2.980 

-6823 

2.748 

.6391 

2.649 

.6207 

2.648 

.6205 

2.367 

-5685 

2.163 

•5320 

i  .067 

.5148 

For  accurate  work,  if  the  temperature  differs  more  than  3°  from  20°, 
a  correction  for  density  of  — o.ooi  for  each  2°  below  20°  or  of  +0.001  for  each 
2°  above  20°  should  be  made. 


1  '  '  ' 

34 

o-2 
J.O 
2& 
•2.Q 
24 
•2.'2 

•>.o 

_  x 

- 

- 

\ 

••N. 

- 

- 

2^ 

- 

- 

\^ 

- 

- 

^ 

x^ 

- 

- 

refractive  IE 
>ity  ;  Rohrbac 
Solution 

\ 

- 

- 

to  dens 

a's 

2 

1.75 


1.70 


1.55 


1.50 


1.65  1.60 

Refractive  Index 
FIG.  722. — Diagram  showing  the  relation  between  refractive  indices  and  density  in  Rohrbach's  solution. 

475.  Molten  Substances  as  Specific  Gravity  Fluids. — Molten  substances 
are  rarely  used  for  the  determination  of  densities,  although  they  may  be  so 
used.     They  are  chiefly  adapted  to  the  separation  of  the  mineral  constituents 
of  a  rock  and  are  described  in  full  below.1 

DETERMINATION  OF  THE  SPECIFIC  GRAVITY  OF  A  MINERAL  WHOSE  DENSITY 
is  GREATER  THAN  THAT  OF  THE  FLUID 

476.  Thoulet  (1879). — If  the  specific  gravity  of  the  mineral  whose  density 
is  to  be  determined  is  greater  than  that  of  the  fluid  in  which  it  is  to  be  im- 
mersed, it  may  be  determined  by  a  method  devised  by  Thoulet.2     In  a 

1  Art.  484. 

2  J.  Thoulet:    Snr  un  nouveau  precede  pour  prendre  la  densite  demineraux  en  fragments 
tres-petits.    Bull.  Soc.  Min.  France,  II  (1879),  189-191. 


536  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  476 

piece  of  wax,  well  smoothed  and  about  the  size  of  a  grain  of  wheat,  is  enclosed, 
as  a  weight,  a  fragment  of  a  mineral  of  such  size  that  the  specific  gravity  of 
the  two  combined  is  between  i  and  2.  The  weight  of  this  sinker  is  repre- 
sented by  G  and  the  weight  of  the  mineral  to  be  determined  by  g.  The  latter 
is  lightly  attached  to  the  wax  by  pressure,  and  the  specific  gravity  of  the 
whole  is  determined  by  inserting  it  in  Thoulet's  solution  diluted  until  the 
mineral  and  wax  neither  sink  nor  float.  Let  the  density  of  the  fluid  at  this 
stage,  and,  consequently,  that  of  the  combined  substances,  be  represented  by 
A.  The  mineral  and  wax  are  now  removed  from  the  solution  and  washed, 
the  adhering  mineral  fragments  carefully  detached,  and  the  density  of  the 
weighted  wax  determined  by  further  dilution  of  the  heavy  solution.  Let 

/^« 

this  value  be  D.     The  volume  of  the  wax,  therefore,  is  V  =  ^;  the  volume  of 

a 

the  mineral,  t>  =  ~v  d  being  the  desired  specific  gravity.     We  have,  therefore: 

<?-}-g  _G+g 

D+d 


CHAPTER  XXXIX 
THE  MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 

477.  Preliminary  Examination. — It  is  necessary  to  separate  a  rock  into 
its  component  minerals  for  such  purposes  as  chemical  analysis,  the  deter- 
mination of  specific  gravity,  or  the  refractive  index  of  a  single  component. 
Preliminary  to  such  separation  by  any  one  of  the  following  methods,  it  is 
necessary  to  break  up  the  rock  fragment.  This  should  be  done,  first  roughly 
in  an  iron-,  then  in  a  diamond-mortar.  The  rock  should  be  crushed, 
not  ground;  the  aim  being  to  separate  all  mineral  particles  from  those  ad- 


FIG.  723. — Preparation    microscope.     About  1/2 
natural  size.     (Fuess.) 


PIG.    724. — Preparation    microscope    after 
Weinschenk.     (Seibert.) 


jacent,  and  to  obtain  as  uniform  a  grain  as  possible.  After  crushing,  the 
dust  and  clay-like  particles  are  removed  by  stirring  the  pulverized  mineral  sev- 
eral times  into  a  beaker  of  water,  and  decanting.  The  residue  is  dried,  and 
sifted  through  a  series  of  metal  sieves  of  varying  mesh.  Ordinarily  a  series 
of  five,  varying  from  i.o  to  0.2  mm.  mesh,  is  sufficient.  Each  grade  of 
powder  is  now  examined  under  a  low  power  microscope  to  determine  whether 
each  grain  consists  of  but  a  single  mineral.  A  very  convenient  form  of  micro- 
scope for  this  purpose  is  that  proposed  by  Kalkowsky1  and  shown  in  Fig.  723. 
The  focus  of  the  lens  A  can  be  adjusted  by  means  of  the  screw  T  for  the 

1  C.  Leiss:  Neues  Lupensiativ  mil  Polarisation  fur  miner  alogische,  geologische  und  palaon- 
tologische  Zwecke.     Neues  Jahrb.,  1897  (I),  81-82. 

Idem:  Die  optischen  Instruments.    Leipzig,  1899,  257-258. 

537 


538  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  478 

examination  of  the  mineral  grains  spread  evenly  over  the  glass  plate  0, 
which  is  8  1/2X10  cm.  in  size  and  set  into  a  brass  frame.  The  two  hinges, 
s  and  si,  permit  the  lens  A  to  cover  every  part  of  this  plate.  Should  it 
be  desired  to  use  polarized  light,  this  is  obtained  by  the  glass-plate  polarizer 
P  and  the  cap  nicol  A . 

Another  form,  proposed  by  Weinschenk,1  is  shown  in  Fig.  724.  In  this 
the  glass  plate  may  be  moved  transversely  across  the  field,  the  box  5  serving 
to  distribute  the  powder  evenly  over  it. 

Having  examined  the  different  powders,  the  largest  grained,  homogeneous 
one  is  taken,  and  the  remaining  coarser  material  is  reduced  to  the  same 
size.  The  larger  the  material,  the  more  readily  is  it  separated  mechanically 
but  the  less  readily  chemically.  Should  the  mineral  be  found  to  be  full  of 
inclusions,  it  will  be  necessary  to  reduce  it  to  still  smaller  size;  and  it  may  be 
necessary  to  use  bolting  cloth  sieves. 

The  microscopic  examination  of  the  powder  will  reveal  the  nature  of  the 
component  minerals.  Should  it  be  intended  to  make  the  separation  by  heavy 
fluids,  such  minerals  as  would  decompose  it  must  first  be  removed.  Most 
of  the  minerals  being  determinable  microscopically,  the  density  of  the 
separating  fluid  for  the  various  separations  will  also  be  known  approximately, 
and  much  time  may  thus  be  saved  in  the  process,  especially  if  a  series  of  fluids 
of  varying  densities  is  on  hand. 

478.  Separation  by  Means  of  the  Electromagnet. — Metallic  iron,  which 
may  have  been  derived  from  the  mortar  in  which  the  rock  was  crushed,  may 
be  separated  from  the  pulverized  rock  by  means  of  an  ordinary  magnet. 
The  removal  of  iron  is  necessary  before  proceeding  to  a  separation  by  means 
of  a  heavy  fluid,  since  many  of  them  are  decomposed  by  metals. 

By  simply  drawing  the  magnet  through  the  powder  repeatedly,  all 
magnetic  particles,  such  as  metallic  iron  and  magnetite,  will  become  attached 
to  it  and  may  be  removed  by  brushing.  They  are  more  easily  removed  if 
the  magnet  is  moved  beneath  the  paper  upon  which  the  powdered  mineral 
lies. 

A  more  convenient  method  was  used  by  Cohen.2  A  sheet  of  smooth 
paper  is  dampened  and  stretched  upon  a  rectangular  frame  so  that  when 
dry  it  is  perfectly  flat.  To  this  frame  are  attached  four  legs  of  such  a 
height  that  the  hand  may  be  used  beneath  it.  The  well-dried  powder  is 
spread  upon  this  frame,  and  the  magnet  is  drawn  underneath  repeatedly, 
from  the  center  toward  the  edge,  carrying  with  it  the  magnetic  particles, 
which  are  swept  off  by  means  of  a  brush  after  removing  the  magnet.  The 

1  E.  Weinschenk:  Die  gesteinsbildenden  Mineralien.     Freiburg  im  Breisgau,  1901,  20. 
English  translation  by  Clark,  New  York,  1912,  161. 

2  E.  Cohen:  Zusammenstellung    petrographischer    Untersuchungsmelhoden.     Stuttgart, 
3  Aufl.,  1896,  12-13. 


ART.  478]   MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS  539 

finer  the  powder,  the  oftener  is  it  necessary  to  repeat  the  operation,  since 
some  of  the  particles  are  held  back  by  the  adjacent  grains. 

Instead  of  a  simple  horseshoe  magnet,  a  magnetized  metallic  graining 
comb,  such  as  is  used  by  painters,  may  be  employed. 

Many  iron-bearing  minerals,  which  in  themselves  are  not  magnetic, 
may  be  separated  from  the  non-iron-bearing  by  a  powerful  electromagnet, 
as  was  first  pointed  out  by  Fouque,1  who  used  it  to  separate  the  ferro-mag- 
nesian  minerals  from  the  feldspars  in  the  lavas  of  Santorin.  The  same 
process  was  used  by  Doelter,2  who  found  that  all  minerals  are  not  attracted 
with  the  same  ease.  In  the  following  table  the  minerals  are  arranged  in 
the  order  of  decreasing  susceptibility.  Those  last  on  the  list  contained 
inclusions. 

1.  Magnetite. 

2.  Hematite,  ilmenite. 

3.  Chromite,  siderite,  almandite. 

4.  Hedenbergite,  ankerite,  limonite. 

5.  Iron-rich  augite,  pleonaste,  arfvedsonite. 

6.  Hornblende,  iron-poor  augite,  epidote,  pyrope. 

7.  Tourmaline,  bronzite,  vesuvianite. 

8.  Staurolite,  actinolite. 

9.  Olivine,  pyrite,  chalcopyrite. 

10.  Biotite,  chlorite,  rutile. 

11.  Hauynite,  diopside,  muscovite. 

12.  Nephelite,  leucite,  dolomite. 

As  may  be  seen  from  the  table,  the  degree  of  attraction  is  not  proportional 
to  the  iron  content,  for  many  iron-rich  minerals,  such  as  pyrite  and  biotite, 
are  less  attracted  than  others  which  are  poorer  in  iron.  Heating  to  redness 
sometimes  increases  the  power  of  being  attracted,  although  the  amount  of 
iron  is  thereby  in  nowise  changed. 

Depending  upon  the  degree  of  attraction,  the  component  minerals  of  a 
rock  may  sometimes  be  separated  by  increasing  or  decreasing  the  strength  of 
the  electromagnet.  This  may  be  done  by  changing  the  strength  of  the 
current,  but  more  readily,  as  suggested  by  Rosenbusch,3  by  decreasing  or 
increasing  the  distance  between  the  poles  of  the  magnet. 

1  F.  Fouque:  Nou-seaux  precedes  d' analyse  mediate  des  roches  et  leur  application  axu 
laves  de  la  dernier e  eruption  de  Santorin.     Comptes  Rendus,  LXXV  (1872),  1089-1091. 

Idem:  Nouveau  procedes  d' analyse  mediate  des  roches  et  leur  application  aux  laves  de  la 
derniere  eruption  de  Santorin.  Mem.  Acad.  France,  XXII  (1874),  n. 

2  C.  Doelter:  Ueber  die  Einwirkttng  des  Elektromagneten  auj  -oerschiedene  Mineralien  und 
seine  Anwendung  behufs  mechanischen   Trennung  derselben.     Sitzb.   Akad.   Wiss.   Wien, 
LXXXV  (1882),  pt.  I,  47-71. 

Idem:  Ueber  die  mechanische  Trennung  der  Mineralien.     Ibidem,  442-449. 
Idem:  Die  Vulkane  der  Capverden,  1882,  72.* 

3  H.  Rosenbusch:  Comment  in  review  of  Doelter,  Op.  cit.,  Neues  Jahrb.,  1882  (II), 
Ref.  252. 


540 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  478 


An  electromagnet  for  separating  minerals  is  shown  in  Fig.  725.  The 
current  is  supplied  by  a  battery  consisting  of  two  strong  accumulators,  or 
one  or  two  strong  chromic  acid  Bunsen  cells.  To  perform  the  separation, 
the  two  L-shaped  soft-iron  poles  are  separated  an  appropriate  distance, 
clamped,  and  the  current  turned  on.  The  pulverized  mineral,  scattered 
over  a  sheet  of  paper,  is  now  brought  beneath  the  magnet,  which  may  pick 
up  a  certain  proportion  of  the  constituents.  When  the  magnet  carries  as 
much  as  it  will  hold,  the  paper  is  removed,  and  a  blank  sheet  is  substituted. 
On  turning  off  the  current  the  mineral  particles  will  drop  to  the  paper. 
The  process  is  repeated  until  no  more  minerals  can  be  extracted  from  the 
powder.  Any  adhering  mineral  particles  are  now  removed  from  the  magnet 


FIG.  725. — Electromagnet.     (Fuess.) 


FIG.  726. — Mann's  electromagnet. 


with  a  brush  or  a  trimmetf  feather,  the  poles  are  moved  closer  together,  and 
another  series  of  mineral  fragments  is  extracted,  and  so  on  until  all  are 
removed. 

The  ordinary  electromagnet  is  adapted  to  the  separation  of  large  quan- 
tities of  iron-rich  constituents  from  a  rock  powder.  If  only  a  small  amount 
of  such  minerals  be  present,  they  may  be  better  separated  by  a  method 
first  proposed  by  Pebal.1  Instead  of  extracting  the  minerals  from  the  dry 

*L.  Pebal:  Ueber  die  Anwendung  -son  Elektromagneten  zur  mechanischen  Scheidung  von 
Mineralien.  Sitzb.  Akad.  Wiss.  Wien,  LXXXV,  Abt.  i,  (1882),  147-148. 

Idem:  Notiz  iiber  mechanische  Scheidung  von  Mineralien.  Ibidem,  LXXXVI  (1882), 
192-194. 


ART.  479]   MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS  541 

powder,  they  are  removed  from  suspension  in  water.  The  powder  is  stirred 
in  a  beaker  of  water,  after  which  a  rod-shaped  electromagnet  is  inserted  and 
with  it  the  stirring  is  continued  until  all  of  the  iron-bearing  minerals  are 
attached. 

An  improvement  on  this  method  was  given  by  Mann,1  who  placed  his 
electromagnet  horizontal  and  fastened  the  two  poles  so  that  their  knife-like 
edges  were  vertical  and  separated  0.5  mm.  Attached  to  this  was  a  burette- 
like  tube  (/,  Fig.  726),  50  cm.  long  and  with  an  inner  diameter  of  6  mm. 
At  the  upper  end  was  a  funnel,  immediately  below  which  a  constriction  in 
the  tube  narrowed  it  to  4  mm.  At  the  lower  end  of  the  tube,  which  ended  a 
few  millimeters  above  the  knife  edges  of  the  poles,  was  a  stop-cock. 

The  mineral  powder  was  stirred  up  with  water  in  a  beaker,  and  was 
poured,  with  one  motion,  into  the  funnel.  This  imprisoned  the  air  within 
the  tube  /,  and  when  the  stop-cock  was  slowly  opened,  the  water  flowed 
between  the  tube  and  the  bubble,  which  practically  filled  the  whole  of  it  and 
prevented  the  clogging  of  the  valve  by  the  settling  of  a  quantity  of  the 
powder.  The  water  and  the  uniformly  distributed  powder  passed  in  a  slow 
stream  over  the  magnet,  which  picked  out  the  iron-bearing  minerals.  When 
enough  were  attached,  the  stop-cock  was  closed,  the  electric  current  shut  off, 
and  the  powder  washed  into  a  beaker.  The  process  was  then  repeated 
until  all  of  the  powder  had  passed  over. 

479.  Separation  by  Means  of  Water. — The  method  of  separating  min- 
erals by  means  of  water  is  not  much  used  in  petrographic  work,  although 
it  is  applicable  in  certain  cases.  It  is  largely  used  in  the  separation  of  the 
constituents  of  soils,  but  that  is  a  different  problem,  and  the  student  is 
referred  to  the  publications  of  the  Bureau  of  Soils.2  It  may  be  used  with 
advantage  in  the  separation  of  heavy  residual  minerals  from  the  decomposi- 
tion products  of  the  rocks,  thus  determining  their  original  igneous  or  sedi- 
mentary character,3  or  even  in  the  separation  of  minerals  from  crushed  un- 
altered rocks.4  The  method  is  simply  the  washing  of  the  material  in  a 

1  Paul  Mann:  Untersuchungen  iiber  die  chemische  Ziisammensetzung  einiger  Augile  aus 
Phonolithen  und  verwandten  Gesleine.     Neues  Jahrb.,  1884  (II),  181-185. 

2  See  also  F.  Steinriede:  Anleilung  zur  miner  alogischen  Bodenanalyse.    Leipzig,  1889.* 
F.  Wahnschaffes :  Anleitung  zur  wissenschaitlichen  Bodenunlersuchung,  Berlin,   1887, 

2  Aufl.,  1903.* 

A.  Nowacki:  Kurze  Anleitung  zur  einfachen  Bodenunter  suchung.     Berlin,  1885.* 
Konrad  Keilhack:  Lehrbuch  der  praktischen  Geologic.     Stuttgart,  1896,  373-389. 

3  Orville  A.  Derby:    On  the  separation  and  study  of  the  heavy  accessories  of  rocks.     Proc. 
Rochester  Acad.  Sci.,  I  (1891),  198-206. 

E.  \V.  Dafert  and  O.  A.  Derby:  On  the  separation  oj  minerals  of  high  specific  gravity. 
Ibidem,  II  (1893),  122-132. 

4  Hans  Thiirach:    Ueber  das  Vorkommen  mikroskopischer  Zirkone  und  Titanminer  alien 
in  den  Gesteinen.    Verhandl.  phys.  med.  Gesell.  Wurzburg,  N.  F.  XVIII  (1884).* 

J.  J.  H.  Teall:  On  the  occurrence  of  rutile  needles  in  clays.  Mineralog.  Mag.,  VII  (1887). 
201-204. 


542  MANUAL  OF  PETROGRAPH1C  METHODS  [ART.  480 

prospector's  pan,  a  process  so  familiar  that  it  hardly  needs  explanation. 
The  material  is  washed  in  water,  running  if  possible,  to  remove  all  of  the  light 
material.  The  pan  is  given  a  rotary  motion,  the  water  is  poured  off,  the 
upper  layers  of  sand,  etc.,  removed  by  the  hand,  more  water  is  added,  another 
swing,  further  removal  of  material,  and  so  on,  until  a  final  rotary  swing  in  a 
very  little  water  sends  the  small  remaining  amount  of  lighter  material 
beyond  the  heavy.  A  final  sorting  may  be  made  by  hand  when  the  minerals 
are  dry.  Calcareous  material,  such  as  shale,  limestone,  or  dolomite,  may 
first  be  treated  with  dilute  hydrochloric  acid,  and  the  residual  insoluble 
material  separated  as  above. 

480.  Separation  by  Means  of  Heavy  Fluids. — One  of  the  simplest  and 
most  useful  of  the  methods  for  separating  minerals  is  that  by  which  they  are 
sorted  according  to  their  specific  gravities  by  means  of  heavy  fluids.    The 
process  simply  consists  in  placing  the  powdered  rock  in  a  vessel  containing 
a  fluid  of  high  specific  gravity,  and  gradually  diluting  the  latter  until  the 
mineral  particles  of  a  certain  density  sink.    These,  with  some  of  the  liquid, 
are  withdrawn  from  the  bottom  of  the  vessel,  a  little  of  the  concentrated 
solution  is  added  to  bring  the  mineral  into  suspension,  and  the  specific 
gravity  of  the  fluid  determined.     The  remaining   minerals  are   separated 
in  the  same  manner  by  the  further  dilution  of  the  heavy  fluid,  and  their 
specific  gravities  are  likewise  determined. 

Preceding  a  separation  by  a  heavy  fluid  there  should  always  be  a  micro- 
scopical examination,  as  mentioned  in  Art.  477.  This  permits  the  removal 
of  such  minerals  as  iron,  calcite,  etc.,  which  act  upon  the  particular  fluid 
used.1  Knowing  the  minerals  contained  in  the  rock,  the  possibility  of 
making  a  specific  gravity  separation  may  be  determined.  An  idea  may  also 
be  obtained  as  to  the  amount  of  dilution  necessary  to  bring  down  certain 
known  minerals,  thus  establishing  the  limits  between  which  the  separation 
must  take  place.  This  may  sometimes  be  of  importance  since  in  the  powder 
there  is  no  way  of  distinguishing  which  mineral  is  falling  and,  if  there  should 
be  present  two  which  have  nearly  equal  densities,  it  is  quite  possible  that 
the  proper  point  of  dilution  might  be  passed. 

481.  Indicators. — Knowing  most  of  the  minerals,  the  separation  limits 
may   be   marked   by   certain  indicators  placed  in   the  fluid.     One   may 
have  to   separate,   for  example,   from   an  olivine  gabbro,  olivine  (3.36), 
augite  (3.36),  hornblende  (3.12),  labradorite   (2.70),  and  magnetite  (5.10). 
The  magnetite  should  first  be  removed  by  means  of  a  magnet.     The  proper 
amount  of  dilution  of  the  heavy  fluid  may  readily  be  found  by  using  as 
indicators,  apatite  (3.18)  and  aragonite  (2.92).    The  indicators  are  placed 
in  the  heavy  solution  with  the  powder  to  be  separated.    When  the  apatite 
sinks  all  of  the  olivine  and  augite  must  have  gone  down;  when  the  aragonite 

1  See  Art.  462. 


ART.  481]    MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 


543 


sinks  there  remains,  on  the  surface  of  the  fluid,  only  the  labradorite.  The 
vessel  containing  the  heavy  fluid  must  be  thoroughly  and  continuously 
shaken  during  the  dilution. 

Goldschmidt1  gives  the  following  table  of  indicators: 


Xo. 

2 

3 
4 

6 

8 

9 
10 
ii 

12 
13 
14 
15 

16 

17 
18 

19 

20 

Indicator 

Locality 

Sp.  gr. 

Difference 

Sulphur 

Girgenti  
Waltsch  

2.070 
2.160 

2.  212 
2.246 
2.284 
2.362 

2-397 
2.465 
2.570 
2.617 
2.650 
2.689 
2-715 
2-733 
2.868 
2.916 

2-933 
3.020 

3-125 
3-i8 

0.090 
0.052 
0.034 
0.038 
0.078 

0-035 
0.068 
0.105 
0.047 
0-033 
0.039 
0.026 
0.018 

o.i35 
0.048 
0.017 
0.087 
0.105 
0.055 

Hyalite 

Opal 

Scheiba 

Xatrolite  
Pitchstone 

Brevig  .... 

Meissen  

Obsidian 

Lipari  Islands  
Hungary  
Vesuvius 

Perlite  
Leucite  
Adularia  
Nephelite  
Quartz  
Labradorite 

St.  Gotthard 

Brevig  .... 

Middleville,  N.  Y  
Labrador  

Calcite  

Rabenstein  

Dolomite  
Dolomite 

Muhrwinkel 

Rauris   .   . 

Prehnite 

Kilpatrick  

Aragonite  
Actinolite  
Andalusite  
Apatite  

Bilin  .      .    . 

Zillerthal  

Bodenmais  

Ehrenfriedersdorf 

As  to  whether  natural  or  artificial  indicators  are  most  desirable,  Gold- 
schmidt2 concluded  that  while  the  former  are  easier  to  recognize  by  their 
colors,  the  latter  are  homogenous  and  may  be  obtained  of  almost  any  specific 
gravity.  He  suggested  that  they  might  be  artificially  colored  or  made  of 
different  forms,  so  that  successive  indicators  could  be  easily  recognized  in 
the  solution.  Later3  he  prepared  sets  of  indicators,  partly  natural  and  partly 
artificial.  One  such  set  comprised  indicators  of  34  different  densities  as 
follows: 


No. 

Sp.  gr. 

No. 

Sp.  gr. 

Xo. 

Sp.  gr. 

Xo. 

Sp.  gr. 

Xo. 

Sp.  gr. 

Xo. 

Sp.  gr. 

2 

3 

4 

2  .060 
2.148 
2.164 
2  .  209 
2252 

7 
8 

9 
10 
ii 

2.311 

2.363 
2.404 
2.448 

2  476 

13 
14 
15 

16 
17 

2-531 
2-552 
2.570 
2.612 

2  646 

19 
20 

21 
22 
23 

2.699 

2  .  72O 
2.740 
2.762 
2  883 

25 
26 
27 
28 

29 

2  .962 
2.981 
3-013 
3-044 

2  os8 

3i 
32 
33 

34 

3-147 
3-189 
3.224 

3-295 

6 

2  298 

12 

2  .492 

18 

2  661 

24 

2  0^6 

^O 

3  °9r 

1  V.  Goldschmidt:    Verh.  k.  k.  geol.  Reichsanst.,  Wien,  1883,  68.* 

Idem:  Ueber    Verwendbarkeit   einer   Kaliumquecksilberjodidlosung   bei   miner  alogischen 
und  peirographischen  Untersuchungen.     Neues  Jahrb.,  B.B.,  I  (1881),  215. 

2  V.  Goldschmidt:    Verh.  k.  k.  geol.  Reichsanst.,  Wien,  1883,.  68.* 

3  Idem:    Ueber  Indikatoren  zur  mechanischen  Gesteinsanalyse  und  speziftschen  Geivichts- 
Bestimmung.     Centralbl.  f.  Min.,  etc.,  1913,  39-44 

These  indicators  may  be  obtained  from  P.  Stoe  in  Heidelberg,  F.  Krantz  in  Bonn,  or 
R.  Fuess  in  Berlin.     The  cost  is  about  25  cents  per  indicator,  sets  being  made  of  various  sizes. 


544 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  482 


Another  set  of  indicators,  given  by  Johnsen  and  Miigge,1  consists  of  cubes 
of  various  kinds  of  glass,  4  mm.  along  edges.  To  make  them  visible  in  the 
solution  they  were  ground  on  five  sides  and  blackened  with  graphite.  On 
the  sixth  side  the  value  of  the  density  was  scratched  with  a  diamond. 

Still  another  set  was  proposed  by  Linck.2  It  consists  of  24  cubes  of 
glass  with  6-mm.  edges  and  ranging  in  specific  gravity  from  2.24  to  3.55, 
with  a  mean  difference  between  two  successive  indicators  of  0.057.  Each 
cube  is  marked  with  a  number  corresponding  to  values  given  in  a  table. 
Linck  suggests  that  a  heavy  solution  containing  indicators  be  stirred,  not  with 
a  glass  rod,  but  with  one  of  hard  rubber,  as  less  likely  to  injure  indicators  or 
beaker. 

482.  TABLE  OF  SPECIFIC  GRAVITIES 

gravities    may   assist   in   choosing   proper    division 


The   following   table   of    specific 
points. 
Cassiterite  ...................... 

Hematite  ....................... 

Magnetite  ....................... 

Monazite  ....................... 

Pyrite  .......................... 

Pseudobrookite  .................. 

Ilmenite  ........................ 

Zircon  .......................... 

Xenotime  ....................... 

Chromite  ....................... 

Rutile  .......................  ... 

Fayalite  ........................ 

Picotite  ......................... 

Perofskite  ....................  ... 

Melanite  ........................ 

Corundum  ...................... 

Siderite  ......................... 

Brookite  ........................ 

Anatase  ......................... 

Orthite  (Allanite)  ................ 

Pleonaste  ....................... 

^Enigmatite  ..................... 

Pyrope  ......................... 

Staurolite  ....................... 

Griinerite  ....................... 

Periclase  ........................ 

Disthene  (Cyanite)  ............... 

Spinel  .......................... 

Grossular  ....................... 

Topaz  ........................... 

Hedenbergite  .................... 

Lavenite  ........................ 


6.65 

5-20 


5-05 
4.98 
4-65 
4-53 
4-52 
4-45 
4-25 
4.14 
4.08 
4-03 
3-95 
3-95 
3-94 
3-94 


85 
.82 
81 

75 
72 


Uwarowite 
Sapphirine 
Rinkite  .  . 


3- 

2 

3 
3 

3. 

3-71 

3-65 

3-62 

3-6o 

3.56 

3-55 

3-54 

3-53 

3-53 

3-5i 

3-49 

3-48 


Titanite 

^Egirite-augite 

Arfvedsonite 

Hypersthene 

Wohlerite 

Barkevikite 

Titan-augite 

Vesuvianite 

Diaspore 

Piedmontite 

Pistacite  (Green  Epidote) 

Augite 

Olivine 

Astrophyllite 

Prismatine 

Jadeite 

Zoisite 

Rosenbuschite 

Diopside 

Bronzite 

Dumortierite 

Axinite 

Diallage 

Cornerupine 

Forsterite 

Sillimanite 

Johnstrupite 

Monticellite 

Fluorite 

Thuringite 

Apatite 

Andalusite 

Anthophyllite 

Humite 

Chondrodite 

Clinohumite . . 


3-48 
3-46 

3-45 
3-45 
3-43 
3-43 
3-4i 
3-4i 
3-40 
3-40 
3-39 
3.36 
3.36 
3-35 
3-34 
3^34 
3-3i 
3-3i 
3-30 
3-29 
3-29 
3.28 

3-27 
3-27 
3-26 


3-15 
3-15 
3-15 
3-15 
3-15 


1  A.    Johnsen   und    O.    Miigge:     Verbesserungen   am   Harada'schen  Trennungsapparat. 
Centralbl.  f.  Min.,  etc.,  1905,  152-153. 

2  G.  Linck:    Indikatoren  zur  Bestimmung  des  spezifischen  Gewichts  von  Flussigkeiten. 
Centralbl.  f.  Min.,  etc.,  1912,  508-509. 

A  set  of  24  indicators  costs  20  Marks.     For  sale  by  F.  Krantz,  Bonn. 


ART.  484]    MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 


545 


Enstatite  ........................     3.15 

Crossite  ........................     3.15 

Spodumen  ......................     3-i5 


Tourmaline 

Hornblende 

Magnesite 

Glaucophane 

Lawsonite 

Lazulite 

Eucolite 

Actinolite  .  . 


3-12 
3-12 


3  -  09 

3  o6 

3-05 

3-02 

Melinophane 3  . 01 

Biotite 3.00 

Gehlenite 3  .  oo 

Tremolite 3  .  oo 

Melilite 3  .  oo 

Mosandrite 2.98 

Datolite 2.95 

Anhydrite 2.95 

Carpholite 2 . 94 

Aragonite 2  . 94 

Eudialyte 2.92 

Dolomite 2 . 90 

Delessite 2  . 89 

Wollastonite '  2  . 88 

Prehnite 2.87 

Muscovite 2.85 

Lepidolite 2.85 

Paragonite 2 . 84 


Phlogopite , 

Pectolite 

Anorthite 

Talc 

Meionite 

B  ytownite 

Pennine  (Chlorite) 

Calcite . . 


81 

2. Si 
2.76 
2-75 
2-73 
2-73 
2-73 
2.71 


2.71 
2.70 
2.70 
2.70 
2.67 
2.65 
2.63 
2.62 
2.62 
2.61 

2.59 
2.58 
2.57 

2-57 
2-55 
2. 55 

2-47 


Clinochlore 

Bastite 

Labradorite 

Alunite 

Andesine 

Quartz. 

Cordierite 

Albite 

Kaolin 

Mizzonite 

Anorthoclase 

Nephelite 

Dipyr 

Serpentine 

Orthoclase 

Microcline 

Leucite 

Brucite 2.39 

Haiiynite 2.38 

Sodalite 2.38 

Hydrargillite 2.38 

Thomsonite 2.35 

Gypsum 2.32 

Tridymite 2 . 30 

Laumontite 2  . 30 

Glauconite 2 . 30 

Scolecite 2.28 

Hydronephelite 2.26 

Epistilbite 2.25 

Natrdlite 2.21 

Opal 

Heulandite 

Phillipsite 

Hyalite 

Analcite 

Stilbite . . 


2  .  21 
2  .  2O 
2  .  20 
2.17 
2.16 
2.16 


Hydromagnesite 2.16 


483.  Heavy  Solutions. — In  the  chapter1  dealing  with  the  determination 
of  the  specific  gravities  of  minerals,  there  were  given  full  descriptions  of 
various  heavy  solutions.     Any  of  the  fluids  there  mentioned  may  be  used 
for  the  mechanical  separation  of  minerals.    They  are,  in  fact,  more  often 
used  in  petrographic  work  for  that  purpose  than  they  are  for  the  determina- 
tion of  specific  gravities. 

484.  Heavy  Melts. — While    the   heavy    liquids    previously    mentioned 
suffice  for  ordinary  separations,  it  occasionally  happens  that  fluids  of  still 
greater  densities  are  needed.    With  the  exception  of  mercury,  there  appears 
to  be  no  liquid  or  solution  with  a  density  greater  than  4.0,  consequently 
recourse  must  be  had  to  melts  of  salts  having  low  fusing  points. 

The  first  heavy  melt  proposed  was  that  of  Breon2  who  used  a  molten 
mixture  of  lead  chloride  (PbCl2)  and  zinc  chloride  (ZnCl2).  The  first  fuses 
at  about  500°  C.  and  has  a  density  of  5.0,  the  second  at  262°  C.  and  has  a 


1  Chapter  XXXVIII,  supra. 

2  R.  Breon:    Separation  des  mineraux  microscoplqiics  lourds. 
Ill  (1880),  46-56. 

35 


Bull.  Soc.  Min.  France., 


546  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  484 

density  of  2.4.  By  the  use  of  various  mixtures  of  the  two  salts,  a  range  of 
from  2.4  to  5.0  is  obtained.  They  are  prepared  by  placing  the  proper  amounts 
of  the  two  substances  in  a  test-tube  on  a  sand-bath  at  about  400°  C.,  and  are 
fused  to  a  homogeneous  mixture.  The  rock  powder,  which  must  not  be  too 
fine  grained,  is  introduced  in  small  installments  with  continuous  stirring 
with  a  platinum  rod.  After  the  heavy  minerals  have  sunk,  the  tube  is  allowed 
to  cool  and  the  salts  to  solidify.  The  mass  is  broken  apart  in  the  middle, 
and  the  separated  mineral  recovered  by  dissolving  the  salt  in  hot  water  to 
which  some  acetic  or  nitric  acid  has  been  added. 

In  1889,  Retgers1  proposed  as  a  heavy  melt,  silver  nitrate  (AgNO3) 
which  melts  at  198°  C.  to  a  clear  transparent  water-like  fluid  with  a  density 
of  4.1.  The  salt  is  fused  in  a  small  beaker  placed  on  wire  gauze  over  an  open 
flame.  Its  density  may  be  reduced  by  the  addition  of  KNO3  ((7  =  2.092)  or 
NaNO3  (6=2.244). 

A  concentrated  solution  of  AgNO3  will  dissolve  a  considerable  amount  of 
silver  iodide  (Agl),  and  soon  a  separation  to  a  yellow  oil-like  and  a  white 
watery  fluid  takes  place.  The  former,  which  may  also  be  obtained  by  melting 
together  the  dry  salts,  has  a  density  of  about  5.0  and  a  melting-point  of  only 
70°  CM  which  makes  it  possible  to  work  over  a  water-bath.  In  composition 
it  is  an  anhydrous  double  salt  of  about  the  composition  2AgNO3+3AgI. 
If  the  Agl  is  in  excess,  the  fluid  is  thick  and  useless,  wherefore  it  is  necessary 
to  use  as  much  AgNO3  as  possible  in  the  combination.  After  separating 
two  minerals,  the  lighter  floating  one  is  removed  with  a  glass  spoon  or  by 
rapidly  pouring  off  the  upper  part  of  the  liquid.  The  adhering  melt  is  re- 
moved from  both  the  light  and  heavy  mineral  grains  by  boiling  in  water. 

In  1893  Retgers2  proposed  the  double  salt,  thallium  silver  nitrate 
(AgTlN2O6).  Although  AgNO3  has  a  melting-point  of  224°  C.  and  T1NO3 
one  of  205°,  the  combination  melts  at  75°.  In  the  above  proportions 
it  has  a  density  of  4.5.3  With  AgNO3  to  T1NO3  in  the  proportion  of  3  to  4, 
it  melts  below  100°  and  has  a  density  of  4.678+.  In  the  proportion  2:4  it 
melts  below  150°  and  has  a  density  of  4.8+.  In  the  proportion  1 14  it  melts 
below  200°  and  has  a  density  of  4.85  +  .  Thallium  nitrate  alone  has  a 
density  of  4.94+  at  the  melting  point. 

1  J.  W.  Retgers:    Ueber  schwere  Flussigkeiten  zur  Trennung  von  Miner  alien.     Neues 
Jahrb.,  1889  ("),  185-192. 

2  Idem:    ThalliumsilbernUrat  als  schwere  Schmelze  zu  Mineraltrennung.     Neues  Jahrb., 
1893  (I),  90-94. 

Idem:  BeitrdgezurKenntniss  deslsomorphismus.  Zeitschr.  f.  physik.  Chem.,  V  (1890), 
451-452,  footnote. 

3  S.  L.  Penfield:    On  some  devices  for  the  separation  of  minerals  of  high  specific  gravity. 
Amer.  Jour.  Sci.,  L  (1895),  446-448. 

Idem:  Ueber  Verbesserungen  der  Methoden  zur  Trennung  von  Miner  alien  mit  hohem 
specifischen  Gewicht.  Zeitschr.  f.  Kryst.,  XXVI  (1896),  134-137.  (Same  paper  as 
preceding.) 


ART.  485]   MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS  547 

The  double  salt  forms  a  fluid,  thin  and  colorless  as  water,  and  is  well 
suited  to  the  separation  of  minerals  except  sulphides  which  it  attacks.  It 
may  be  diluted  with  water  in  any  proportion. 

Retgers,1  in  1896,  made  many  experiments  on  heavy  melts  to  obtain 
one  which  would  not  attack  sulphides  and  yet  be  of  high  specific  gravity. 
With  acetates  he  obtained  rather  unsatisfactory  results,  the  specific  gravities 
in  general  being  too  low,  seldom  exceeding  4.0.  The  most  satisfactory  one 
was  thallium-nitrate-acetate  (TlAc+TlNO3)  which  melts  at  65°  (TIAc, 
110°  C;  TINOa,  205°),  has  a  density  of  approximately  4.5,  and  does  not  act 
upon  sulphides.  It  is,  however,  readily  decomposed  at  high  temperatures. 
Below  100°  it  is  stable. 

Among  the  nitrates  several  appear  to  be  serviceable.  Mercuro-nitrate 
(HgNOa+aq)  has  a  density  of  4.3  and  a  melting-point  of  70°  C.  It  is  clear, 
mobile,  miscible  in  all  proportions  with  water,  and  cheap.  It  is  decomposed 
during  heating  but  this  is  of  little  consequence  if  one  works  rapidly.  Thallium 
mercuri-nitrate  (TlHg(NO3)4)  has  a  density  of  5.0  and  a  melting-point  of 
110°  C.  It  is  a  mobile  fluid,  miscible  in  all  proportions  with  water,  does  not 
act  on  sulphides,  but  is  rather  cloudy  and  therefore  not  so  useful  as  thallium 
mercuro-nitrate  (TlHgN2Oe).  This  has  a  density  of  5.3  and  a  melting-point 
of  76°  C.  It  is  a  transparent  mobile  fluid,  miscible  in  all  proportions  with 
water,  and  does  not  act  on  the  sulphides  of  the  metals.  It  is  not  decomposed 
below  100°  C.  It  may  be  prepared  by  adding  crystals  of  mercuro-nitrate 
(HgNOs+aq)  to  an  equal  quantity  of  molten  thallium  nitrate.  It  forms  a 
double  salt  with  a  melting-point  much  lower  than  that  of  either  component. 
The  fusion  can  be  performed  in  a  test-tube  over  an  open  flame,  but  care 
must  be  used  not  to  fill  the  tube  too  full  since  the  material  boils  up  consider- 
ably, due,  probably,  to  the  escape  of  the  water  from  the  mercuro-nitrate. 
Should  the  temperature  be  raised  too  high,  brown  vapor  passes  off  but  this 
ceases  when  the  heat  is  reduced.  On  cooling,  the  pale  yellow  fluid  becomes 
cloudy  and  viscous  at  85°,  andsoldifies  at  74°  .  To  make  separations  with 
this  fluid,  one  should  work  over  a  water-bath. 

SEPARATING  APPARATUS 

485.  Thoulet  (1879). — The  first  apparatus  used  for  the  separation  of 
minerals  by  means  of  heavy  solutions  was  that  proposed  by  Thoulet2  (Fig. 
727).  This  consists  of  a  graduated  tube,  closed  at  the  lower  end  by  a  pair 

1  J.  W.  Retgers:     Versuche  zur  Darstellung  neiier  schiverer  Flussigkeiten  zur  Miner  al- 
trennung.     I.  Die  Acetate  der  Schwermelalle  als  schwere  Schmelzen.     Neues  Jahrb.,  1896  (i), 
212-221. 

Idem:  Versuche  usw.  II.  Die  Nitrate  und  Doppelnitrate  der  Schu<ermetalle  als  schwere 
Schmelzen.  Neues  Jahrb.,  1896  (II),  183-195. 

2  J.  Thoulet:    Separation  mechanique des  divers  elements  miner alogiques  des  roches.    Bull. 
Soc.  Min.  France,  II  (1879),  17-24. 


548  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  486 

of  large- way  stop- cocks  C  and  D.  About  60  c.c.  of  the  heavy  solution  are 
placed  in  the  tube,  and  from  i  to  2  grm.  of  the  pulverized  rock,  previously 
boiled  with  water  to  free  it  from  air,  are  slowly  added  from  above.  The 
upper  end  of  the  tube  is  closed  by  a  stopper  E  through  which  passes  a  tube  F 
connected  with  an  air  pump.  The  stop-cock  C  is  carefully  opened,  and  at 
the  same  time  a  continuous  current  of  air  is  admitted  through  B.  This  serves 
to  keep  the  fluid  and  pulverized  mineral  in  commotion  and  intimately  mixes 
them,  and  also  prevents  the  formation  of  dry  lumps.  No  fluid  should  enter  B 
until  the  mixing  is  complete.  The  cock  C  is  now  closed  and  all  minerals 
heavier  than  the  solution  fall  to  the  bottom  of  the  tube  A .  When  all  have 
^_^^  settled,  the  valve  C  is  opened  and  the  powder  falls  into  the 

g  space  between  C  and  D.  The  valve  C  is  now  closed,  the  end 
(H)  of  the  tube  is  placed  in  a  beaker  of  water,  and  the  mineral 
allowed  to  fall  into  it  by  opening  the  cock  D.  The  space  between 
C  and  D  may  be  rinsed  by  repeatedly  sucking  the  water  as  high 
as  necessary  into  the  tube  B  and  allowing  it  to  fall  back  into 
the  beaker. 

The  dilute  solution  in  the  beaker  is  now  poured  off,  the 
mineral  is  repeatedly  washed,  and  finally  dried.  On  account  of 
the  cost  of  Sonstadt's  solution,  the  dilute  solution  which  was 
poured  off,  as  well  as  all  of  the  wash  water,  is  concentrated  over 
a  water-bath,  and  used  again  and  again. 

Water  is  now  carefully  added,  drop  by  drop,  and  air  is  blown 
through  the  solution  in  A  until  minerals  of  a  less  density  fall. 
The  process  is  repeated  as  before  and  so  on,  again  and  again, 
727.—  until  all  of  the  powder  has  come  down  in  installments  of  different 

Th°UratusaPPa"  sPecific  gravities. 

The  Thoulet  apparatus  is  complicated  and  easily  broken. 
The  proper  control  of  the  valve  C,  to  prevent  the  mineral  from  falling  below 
the  entrance  of  B  before  it  is  thoroughly  separated  from  other  particles,  is 
difficult.  There  is  also  a  likelihood  of  choking  the  narrow  tube  above  C,  and 
of  particles  adhering  just  above  the  constriction  and  coming  down  later 
with  minerals  of  less  density. 

486.  Goldschmidt  (1881). — To  overcome  the  difficulties  attending  the 
use  of  the  Thoulet  apparatus,  and  especially  to  make  possible  a  stirring 
together  of  powder  and  fluid,  Goldschmidt1  preferred  using,  especially  for 
preliminary  separations,  small,  slender,  lipped-breakers  with  a  capacity  of 
from  40  to  50  c.c.  By  their  use  a  much  larger  quantity  of  material  may  be 
worked  and  the  density  at  any  time  may  be  tested  readily.  Further,  no 
special  apparatus  is  required,  and  there  is  no  difficulty  in  cleaning  up. 

1  V.  Goldschmidt:  Ueber  Verwendbarkeit  einer  Kaliumquecksilberlosung  bel  mineralog- 
ischen  und  pelrographischen  Untersuchungen.  Neues  Jahrb.,  B.B.,  I  (1881),  214-261. 


ART.  487]   MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 


549 


Goldschmidt's  method  consists  in  placing  the  pulverized  mineral  in  a 
beaker  with  approximately  30  c.c.  of  the  solution,  stirring  vigorously  with 
a  glass  rod,  allowing  it  to  settle,  and  pouring  off  some  of  the  fluid  with  the 
floating  minerals.  If  the  quantity  is  large,  most  of  the  floating  minerals 
may  be  removed  with  a  glass  or  platinum  spoon.  Upon  pouring  off  the 
solution,  there  will  remain,  hanging  to  the  sides  of  the  vessel,  a  partial  ring 
of  the  lighter  minerals,  broken  on  the  side  where  the  liquid  was  poured  out. 
This  clear  passageway  may  be  widened  somewhat  by  wiping  with  the  glass 
rod,  and  along  it  the  heavy  minerals  may  be  poured  or  rinsed  into  another 
beaker.  Finally  the  adhering  lighter  minerals  may  be  washed  into  a  beaker, 
dried,  and  further  separated.  To  obtain  perfect  separa- 
tion the  process  should  be  repeated  with  each  portion  of 
different  density  thus  obtained. 

487.  Harada  (1881). — One  of  the  simplest  and  most  con- 
venient of  separating  instruments,  and  one  after  which  many 
others  are  patterned,  is  that  designed  by  T.  Harada1  (Fig. 
728).  It  consists  of  a  long,  pear-shaped  glass  vessel,  closed 
at  the  upper  end  by  a  tight  fitting  stopper,  and  at  the  lower 
by  a  stop-cock  whose  passage-way  is  of  the  exact  size  of 
the  inner  diameter-  of  the  tube,  thus  leaving  no  shoulder. 
The  mixing  of  powder  and  fluid  is  carried  out  by  shaking, 
after  which  the  heavy  portion  is  allowed  to  settle,  the 
lower  end  of  the  tube  is  placed  in  the  bottom  of  the  vessel 
b,  the  cock  is  carefully  opened,  and  the  mineral  particles  are 
permitted  to  pass  through.  A  portion  of  the  fluid,  only 
enough  to  cover  the  opening  of  the  tube,  will  also  escape, 
but  most  of  it  will  be  held  in  the  vessel  by  the  air  pres-  FlG'  72t8u^Harada 
sure.  After  all  of  the  mineral  has  passed  out,  the  stop-cock 
is  closed  and  a  thin  layer  of  water  is  poured  over  the  solution  in  the  vessel 
b.  Upon  raising  the  tube  slightly,  the  water  will  rush  into  the  narrow  exten- 
sion while  the  heavy  solution  will  fall  and  carry  with  it  any  remaining  particles. 

The  separating  operation  is  repeated  with  the  addition  of  a  small  amount 
of  water  to  permit  minerals  of  lesser  density  to  fall.  If  perfectly  pure 
material  is  desired,  it  is  necessary  to  perform  the  separation  a  number  of 
times,  since  the  heavier  particles  mechanically  carry  down  with  them  some 
of  the  lighter  and  the  lighter  hold  up  part  of  the  heavier. 

Johnsen  and  Mugge2  suggest  that  the  receiving  cup  be  made  cylindrical 
and  very  little  larger  than  the  lower  portion  of  the  tube.  If  indicators3 

1  K.  Oebbeke:    Beilrage  zur  Petrographie  der  Philippinen  und  der  Palau-Inseln.     Neues 
Jahrb.,  B.B.,  I  (1881),  457. 

2  A.    Johnsen    und   O.    Mugge:    Verbesserungen   am  Harada' schen    Trennungsapparat. 
Centralbl.  f.  Min.,  etc.,  1905,  152-153. 

3  Art.  480. 


550 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  488 


are  used,  they  will  pass  with  the  mineral  grains  freely  through  the  stop-cock 
below,  and  the  density  of  the  mineral  may  at  once  be  determined  from  the 
figures  inscribed  on  the  accompanying  glass  cube.  Johnsen  and  Miigge 
also  suggest  adding  the  water  necessary  for  dilution  as  well  as  the  indicators 
at  the  lower  end  of  the  tube,  inverting  it  for  this  purpose.  The  upper  stopper 
need  not  be  removed  except  for  the  insertion  of  the  mineral  powder  and  for 
cleaning  purposes. 

488.  Oebbeke    (1881). — Oebbeke1    designed    a   simplified  form  of  the 
Thoulet  tube  which  possesses  all  of  the  good  qualities  of  the  latter,  but  is 
more  readily  cleaned  and  is  much  easier  to  handle. 
Through  a  glass  tube  (Fig.  729),  extending  through 
the  stopper  nearly  to  the  bottom  of  the  vessel,  a  cur- 
rent of  air  is  blown  which  thoroughly  mixes  the  fluid 
and  the  powder.     The  space  between  the  two  stop- 
cocks is  cleaned  by  means  of  a  stream  of  water  forced 
through  the  tube  b,  which  is  bent  and  drawn  out  to 
a  fine  end. 


I]  A' 


FIG.  729- — Oebbeke  apparatus. 


FIG.  730. — Separating 
funnel.  1/4  natural  size. 
(Fuess.) 


489.  Van  Werveke  (1883). — A  very  simple  separating  funnel  was  suggested 
by  Van  Werveke2  in  1883.  It  consists  of  a  funnel  (Fig.  730)  with  a  stop- 
cock having  an  opening  of  the  exact  size  of  the  bore  of  the  tube  to  prevent  the 
lodging  of  any  mineral  particles.  If  such  a  funnel  is  not  at  hand,  the  tube 
of  an  ordinary  funnel  may  be  cut  to  a  convenient  length  and  a  rubbler  tube 

1  K.  Oebbeke:    Op.  cit,  p.  456. 

2  L.  van  Werveke:    Ueber  Regeneration  der  Kaliumquecksilberjodid  Losung  und  iiber  einen 
einfachen  Apparal  zur  Trennung  mittelst  dieser  Losung.     Neues  Jahrb.,  1883  (H),  86-87. 


ART.  490]   MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 


551 


and  pinch-cock  attached.  The  mineral  powder  and  heavy  fluid  are  mixed 
in  the  funnel  by  means  of  a  glass  rod,  or  the  upper  rim  of  the  funnel  is  ground, 
covered  by  a  ground  glass  plate,  and  the  apparatus,  with  the  contained  fluid,  is 
shaken  thoroughly. 

490.  Brogger  (1884). — To  avoid  the  necessity  of  performing  the  operation 
of  separation  a  number  of  times  to  obtain  pure  material,  as  is  necessary 
with  the  Harada  tube,  Brogger1  devised  the  repeating  separator  shown  in 
Figs.  731-733.  This  instrument  differs  from  the  Harada  tube  in  having  a 
second  large  stop-cock  above  the  first.  The  heavy  solution  and  powdered 


FIG.  731.  FIG.  732.  FIG.  733. 

FIGS.  731  to  733. — Brogger 's  method  for  separating  minerals. 

rock  are  placed  in  the  tube  with  the  valve  A  open.  It  is  thoroughly  shaken 
and  the  minerals  are  allowed  to  settle,  as  shown  in  Fig.  731.  The  grains 
now  will  be  in  the  same  incomplete  state  of  separation  as  when  the  Harada 
tube  is  used.  Above  the  lower  valve  will  lie  the  greater  part  of  the  heavy 
mineral  (H)  mingled  with  some  of  the  lighter  (L')  while  floating  on  top  will 
be  a  little  of  the  heavy  (H')  supported  by  the  light  (L).  The  valve  A  is 
now  closed,  the  instrument  is  inverted,  again  thorougly  shaken,  and  set  away 
in  this  position  for  an  hour  (Fig.  732).  Between  the  stopper  6"  and  the  valve 
A  and  between  the  two  valves,  a  further  separation  of  the  constituents  has 
taken  place  as  shown  in  the  figure.  The  instrument  is  now  carefully 
turned  to  the  position  shown  in  Fig.  733,  and  the  valve  A  is  slowly  opened, 
whereby  the  separated  portions  of  the  light  and  the  heavy  minerals  unite  by 
the  movement  shown  by  the  arrows.  After  placing  the  end  of  the  apparatus 

1  W.  C.  Brogger:    Om  en  ny  konstruktion  aj  et  isolations-apparat  for  pelrografiske  under- 
sogelser.     Geol.  Foren.  i  Stockholm  Forh.,  VII  (1884),  417-427. 
E.  Cohen:  Review  in  Neues  Jahrb.,  1885  (i),  395-396. 


552 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  491 


in  a  vessel  whose  cup  is  narrow  at  the  bottom,  the  lower  valve  may  be  opened 
and  the  heavy  mineral  removed  as  from  the  Harada  tube.  After  the  valve 
B  is  closed  the  process  is  repeated  until  no  further  separation  takes  place. 
For  the  remaining  minerals  of  less  density  the  solution  is  diluted  and  the 
process  repeated. 

491.  Smeeth  (1888). — The  apparatus  constructed  by  Smeeth1  possesses 
so  many  good  qualities  that  it  seems  strange  that  it  has  not  been  more 
commonly  used.  It  contains  no  stop-cocks  to  leak  or  get  clogged,  and  is 


O       Q 


0 


PIG.  734- 


FIG.  735. 


PIG.  737- 


^.  734.  riLr.   735.  j.- IV*.    tju.  no.  737. 

FIGS.  734  to  736. — Smeeth  separating  apparatus.     FIG.  737. — Diller's  modification  of  the  Smeeth 

apparatus. 

easily  cleaned.  It  consists,  essentially,  of  a  turnip-shaped  separating  vessel, 
closed  above  and  below  by  ground  glass  stoppers  (Fig.  734).  The  lower 
stopper  (Fig.  735)  is  in  the  form  of  a  glass  rod  with  enlarged  end,  and  the 
whole  apparatus  fits  tightly  in  a  flat,  candle-stick-shaped  foot. 

The  separation  is  begun  with  the  inner  rod  removed  and  the  apparatus 
set  up  as  shown  in  Fig.  734.  On  closing  the  upper  stopper,  the  whole 
instrument  may  be  vigorously  shaken.  After  the  heavy  mineral  settles, 
the  stopper  (Fig.  735)  is  inserted,  and  the  whole  upper  part  (Fig.  736)  is 
removed.  By  using  several  cup-bases,  the  minerals  of  different  densities 
may  be  separated  without  interruption.  Afterward  the  densities  of  the  fluids 
contained  in  the  cups  may  be  determined,  without  removal,  by  the  Westphal 
balance. 


1  W.  F.  Smeeth:    An  apparatus  for  separating  the  mineral  constituents  of  rocks. 
Roy.  Soc.  Dublin,  VI  (1888),  58-60. 


Proc. 


ART.  494]    MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS  553 

492.  Diller  (1896). — The  Smeeth  apparatus  was  improved  by   Diller1 
who  gave  it  a  pear-  rather  than  a  turnip-shape  (Fig.  737).     It  thus  offers  less 
chance  for  the  lodgment  of  particles  on  the  sides.     By  making  the  upper 
stopper  hollow  and  shortening  the  rod-stopper,  the  apparatus  may  be  shaken 
with  the  latter  in  place  and  a  second  partial  separation  made.     By  tilting  the 
instrument  in  a  manner  similar  to  that  described  for  the  Brogger  tube,  and 
then  removing  the  rod,  repeated  separations  may  be  made  very  easily. 

493.  Laspeyres  (1896;. — A  simple  and  cheap  instrument,  proposed  by 
Laspeyres,2  is  shown  in  Fig.  738.     It  may  be  made  of  any  size,  the  most 
convenient  being  one  which  requires  but  12  c.c.  of  fluid  and  may  be  used  with 
one  hand.     It  consists  of  two  pear-shaped  bodies  connected  at 

their  small  ends  by  the  stop-cock  C,  whose  opening  is  equal  to 
that  of  the  inner  diameter  of  the  tube  at  that  point.  At  each 
end  are  large,  carefully  ground  stoppers.  By  placing  the  thumb 
and  two  fingers  in  the  depressed  ends  of  the  stop-cock  and 
the  stoppers,  a  firm  grip  may  be  obtained  upon  the  instru- 
ment, thus  permitting  vigorous  shaking  with  no  danger  of 
losing  the  enclosed  liquid.  To  use  the  instrument,  the  cock  C 
is  opened  and  the  double  containers  filled  as  full  as  possible  with 
the  powdered  rock  and  separating  liquid.  The  end  is  closed 
and  the  vessel  is  thoroughly  shaken,  producing  the  first  partial 
separation.  After  permitting  the  minerals  to  settle,  the  valve  PIG.  738.— 

is  closed  and  the  shaking  repeated.     When  the  valve  is  again  LasPeyres  sep- 
arating   appa- 
opened,  the  minerals  of  the  second  separation  unite  with  those  ratus. 

of  the  first,  as  in  previously  described  methods 

A  somewhat  similar  instrument  is  described  by  Hauenschild.3 

494.  Wulfing  (1890). — Another  repeating  separator,  based  somewhat  on 
Brogger's  principle  but  of  quite  different  form,  is  that  proposed  by  Wulfing. 4 
Instead  of  being  pear-  or  turnip-shaped  this  one  is  link-like  (Fig.  739).     At 
the  upper  and  lower  ends  are  valves  by  which  the  connection  between  the 
two  parts  may  be  shut  off.     When  open,  the  bore  is  the  same  as  the  tube 
adjacent,  so  that  there  is  no  chance  for  the  collection  of  mineral  particles  at 
this  point.     The  capacity  of  the  instrument  is  about  40  c.c.     It  is  filled, 
through  two  glass-stoppered  openings,  about  three-fourths  full  of  the  heavy 
fluids,  the  connecting  valves  being  opened  to  bring  it  to  the  same  level  on 
either  side.     The  powder  is  then  added  to  both  sides,  the  connecting  valves 
are  closed,  and  the  first  partial  separation  takes  place  as  in  the   Brogger 

1  J.  S.  Diller:    The  Smeeth  separating  apparatus.     Science,  N.  S.,  Ill  (1896),  857-858. 

2  H.  Laspeyres:    Vorrichtung  zur  Scheidung  von  Miner  alien  mittelst  schwerer  Losungen. 
Zeitschr.  f.  Kryst.,  XXVII  (1896),  44-45. 

3  Alb.  Hauenschild:  Zeitschr.  f.  Baumaterialienkunde,  1898.* 

4  E.  A.  Wulfing:    Beitrag  zur  Kenntniss  des  Kryokonit.     Neues  Jahrb.,  B.B.,  VII  (1890), 
164-165. 


554 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  495 


instrument.  Upon  opening  the  lower  valve  and  inclining  the  apparatus, 
the  heavy  material,  with  the  small  amount  of  light  material  which  is  carried 
down  with  it,  will  sink  into  the  lower  tube.  If,  in  the  inclined  position,  the 
upper  valve  is  opened  quickly,  the  difference  in  pressure  will  carry  every 
particle  of  the  heavy  material  down.  The  inclination  of  the  apparatus  is 
now  increased  until  the  liquid  in  the  lower  tube  reaches  the  upper  valve, 
when  the  lower  valve  is  closed.  By  slightly  shaking,  the 
light  material  can  be  washed  over  into  the  now  partially 
emptied  tube.  Should  some  still  remain  in  the  first  tube, 
the  instrument  need  only  be  once  more  inclined,  the  lower 
tube  again  filled  to  the  upper  valve,  and  the  light  powder 
shaken  over.  With  both  stop-cocks  closed,  the  instru- 
ment may  be  shaken  again,  after  wrhich  the  separation  may 
be  repeated,  again  and  again,  in  the  same  manner  as  before, 
until  all  of  the  material  is  separated  according  to  densities. 
The  heavy  material  may  now  be  removed  through  one  of 
FIG.  739.— Wuifing's  the  glass- stoppered  openings,  more  fluid  added,  diluted  to 

separator.      (Puess.)  ^r 

the  proper  density,  and  the  separation  continued  as  with 
other  instruments. 

495.  Luedecke  (1911). — An  apparatus  similar  to  that  of  Smeeth  was  made 
by  Leudecke  and,  after  his  death,  described  by  Dreibrodt.1     It  differs  from 


FIG.  740. 


FIG.  741- 


FIGS.  740  and  741. — Leudecke  apparatus. 

the  Smeeth  apparatus  chiefly  in  the  form  of  the  upper  vessel,  which  is  here 
covered  with  a  ground  glass  plate  (E,  Fig.  740),  and  in  the  stopper  G  (Fig. 
741),  which  is  provided  with  a  capillary  tube  H  to  take  up  the  displaced  solu- 
tion when  it  is  inserted  in  it. 

496.  Separation  Apparatus  for  Heavy  Melts. — To  simplify  separation  by 

1  O.   Dreibrodt:    Trennungsapparat  nach  Prof.  Dr.  0.  Luedecke.     Centralbl.  f.  Min., 
etc.,  1911,425-426. 


ART.  496]    MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS 


555 


means  of  melts,  Penfield1  proposed  the  apparatus  shown  in  Fig.  744.  In  its 
first  form  (Fig.  742)  it  consisted  of  a  glass  tube  with  tapering  end,  into  which 
a  glass  rod  was  ground  with  fine  emery  to  serve  as  a  stop-cock.  The  appara- 
tus was  placed  inside  a  test-tube  and  was  so  adjusted  that  it  reached  to 
within  a  few  millimeters  of  the  bottom.  Test-tube  and  contained  separating 


FIG.  742.         FIG.  743.  FIG.  744. 

FIGS.  742  to  744. — Penfield  separating  apparatus  for  heavy  melts. 

apparatus  were  placed  in  a  beaker  of  hot  water  and  were  heated  until  the 
double  nitrate  of  silver  and  thallium,  which  was  used  in  the  separation,  were 
fused.  A  glass  rod,  bent  as  shown  in  Fig.  743,  was  used  as  a  stirring  rod. 
When  the  separation  had  taken  place,  the  inner  rod  was  raised  and  the 
heavy  mineral  and  a  certain  amount  of  the  melt  were  allowed  to  escape  into 
the  test-tube. 

In  its  improved  form2  the  instrument  is  shown  in  Fig.  744.     The  glass 

1  S.  L.  Penfield  and  D.  A.  Kreider:    On  the  separation  oj  minerals  of  high  specific  gravity 
by  means  of  the  fused  double  nitrate  of  silver  and  thallium.     Amer.  Jour.  Sci.,  XL  VIII  (1894), 
143-144. 

2  S.  L.  Penfield  :  On  some  devices  for  the  separation  of  minerals  of  high  specific  gravity. 
Amer.  Jour.  Sci.,  L  (1895),  446-448. 

Idem:  Ueber  einige  Verbesserungen  der  Methoden  zur  Trennung  von  Miner  alien  mit 
hohem  specifischen  Gewicht.  Zeitschr.  f.  Kryst.,  XXVI  (1896),  134-137.  (Translation  of 
preceding.) 


556  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  497 

tube  b  is  about  20  cm.  long  and  has  an  internal  diameter  of  2.2.  cm.  At  its 
lower  end,  with  well-ground  joints,  it  fits  the  cap  c  and  the  hollow  stopper  a. 
The  whole  apparatus  is  placed  within  a  test-tube  d,  and  is  heated  in  a  beaker 
of  water. 

In  making  a  separation  by  means  of  a  heavy  melt,  Penfield  recommended 
first  melting  the  double  salt  in  a  dish  on  the  water-bath  until  it  becomes 
perfectly  clear.  It  is  then  diluted  with  water  until  it  is  of  approximately  the 
specific  gravity  of  the  mineral  to  be  separated,  and  is  poured  into  the  warm 
separating  apparatus  after  removing  the  stopper  a.  The  tube  b  is  filled  about 
half  full,  after  which  the  mineral  powder  is  added  and  thoroughly  mixed  by 
blowing  a  stream  of  air  into  the  liquid  through  a  small  glass  tube.  Successive 
portions  of  water  are  added  until  the  desired  separation  takes  place.  To 
obtain  the  heavy  material  in  c,  the  previously  warmed  stopper  a  is  inserted, 
the  cap  c  removed,  and  the  contents  washed  out  with  hot  water.  The  cap 
c,  or  one  similar,  may  be  replaced,  more  water  added,  and  another  separation 
made.  For  large  quantities  of  material  the  larger  cap  c'  may  be  used,  though 
it  is  not  so  convenient  as  the  smaller  one. 

497.  Causes  Likely  to  Produce  Errors  in  Separating  Minerals  or  in 
Determining  Specific  Gravities  by  Means  of  Heavy  Fluids. — The  causes 
which  operate  to  produce  wrong  determinations  of  the  specific  gravity, 
or  incomplete  separation,  by  means  of  heavy  solutions,  are:  (a)  The 
variations  in  the  specific  gravity  of  grains  of  the  same  mineral.  This 
may  be  caused  by  inclusions  of  bubbles  which  decrease  the  apparent 
density,  or  by  inclusions  of  other  minerals  which  increase  or  decrease  it, 
or  by  partial  alteration  or  weathering,  (b)  The  impossibility  of  completely 
breaking  apart  the  different  mineral  grains  by  crushing,  wherefore  certain 
grains  may  be  composed  of  more  than  one  mineral.  This  is  especially  the 
case  with  minerals  intimately  intergrown,  as  microperthite,  etc.,  (c)  The 
practical  identity  of  the  specific  gravities  of  certain  minerals,  (d)  The  likeli- 
hood of  the  heavy  minerals  carrying  down  the  lighter,  and  vice  versa,  (e)  The 
consolidation  of  the  grains  into  little  lumps. 1 

A  preliminary  examination  under  the  microscope  will  act  as  a  check  on 
a  and  b,  while  repeated  separation  will  eliminate  d,  the  purity  of  the  material 
being  determined  microscopically.  In  the  third  case,  c,  the  separation  may 
be  made  with  great  care  when  the  densities  are  not  quite  the  same,  the 
microscope  in  this  case  also  showing  when  the  separation  is  complete.  If 
the  densities  are  exactly  equal,  they  cannot,  of  course,  be  separated  by  this 
method.  To  prevent  the  mineral  grains  from  gathering  together  into  little 
dry  lumps  (e)  which  are  afterward  almost  impossible  to  separate,  one  should 

1  If  the  material  is  being  separated  for  chemical  analysis,  one  must  be  certain  that  no 
chemical  change  has  taken  place.  See  W.  F.  Hillebrand:  A  danger  to  be  guarded  against 
in  making  mineral  separations  by  means  of  heavy  solutions.  Amer.  Jour.  Sci.,  XXXV 
(1913),  439-440- 


ART.  499]   MECHANICAL  SEPARATION  OF  ROCK  CONSTITUENTS  557 

first  pour  over  the  powder  only  just  enough  of  the  heavy  solution  to  cover 
it,  and  then  stir  with  a  glass  rod  until  all  of  the  grains  are  thoroughly  wet, 
after  which  more  fluid  may  be  added.  If  the  mineral  was  powdered  some 
time  previously,  and  it  has  been  exposed  to  the  air,  it  is  usually  advisable  to 
boil  it  or  place  it  under  an  air  pump  before  performing  the  separation. 

498.  Separation  of  Thin  Flakes  and  Fine  Needles. — Micaceous  minerals 
may  be  readily  separated  from  a  rock  powder  by  means  of  water.     If  the 
powdered  rock  is  placed  in  a  beaker,  and  a  fine  stream  of  water  is  conducted 
to  the  bottom  by  means  of  a  glass  or  rubber  tube,  the  flakes  will  be  carried 
upward  by  the  eddies  and  thus  be  washed  over  the  edge  of  the  beaker  into 
a  larger  one  surrounding  it. 

Rosenbusch1  separated  mica  from  a  mica  trachyte  by  letting  the  powder 
glide  many  times  down  a  slightly  inclined  sheet  of  rather  rough  writing 
paper.  The  mica  particles  adhered  to  the  paper  while  the  feldspar  slipped 
down. 

Linck2  allowed  small  portions  of  rock  powder  to  fall  from  a  considerable 
height  into  a  glass  funnel  upon  the  inner  surface  of  which  a  slight  coating  of 
moisture  had  been  placed  by  breathing  upon  it.  The  mica  flakes  adhered 
to  the  funnel  while  the  rounded  grains  fell  through.  By  tapping  the  funnel 
lightly  upon  a  sheet  of  paper  on  a  table,  the  mica  dropped  down.  The  process 
was  repeated  several  times  to  obtain  pure  material. 

499.  Separation  by  Hand. — In  spite  of  the  most  careful  separation  or 
repeated  separations  by  heavy  solutions  or  electromagnet,  it  often  will  be 
found,  upon  examination  under  the  microscope,  that  the  constituents  thus 
obtained  are  not  perfectly  homogeneous.     It  may  be  that  the  specific  gravity 
of  two  minerals  is  so  nearly  the  same  that  they  come  down  together,  or  that 
occasional  grains  in  an  otherwise  homogeneous  powder  are  composed  of  two 
minerals  grown  together.     Perhaps  it  is  desired  to  isolate  certain  grains 
before  making  a  separation  by  heavy  solutions,  or  to  pick  certain  constitu- 
ents from  a  sand.     In  all  such  cases  the  only  method  is  to  perform  the 
operation  by  hand.     The  mineral  grains  are  strewn  on  the  stage  of  a  prepa- 
ration microscope   (Figs.  723-724),  and   are  picked  out  by  means  of   the 
moistened  end  of  a  pointed  piece  of  wood  or  the  end  of  a  heavily  waxed 
thread.3    Zirkel4  suggested  picking  out  the  grains  by  means  of  a  dissecting 
needle  upon  whose  point  a  trace  of  Canada  balsam  had  been  placed.     As 
soon  as  the  drop  of  balsam  is  covered  everywhere  with  mineral  grains,  the 

1  H.  Rosenbusch:  Glimmer tr achy t  von  Monlecatini  in  Toscana.  Neues  Jahrb.,  1880  (n)> 
207. 

G.  Linck:    Abhandl.  zur  geolog.  Spez.-Karte  von  Elsass-Lothringen.     Ill,  Strassburg, 
1884,  41-42.* 

3  Rosenbusch- Wiilfing:    Mikroskopische  Physiographic,   Stuttgart,   4   Aufl.,   Ii,    1904, 

434-435- 

4  F.  Zirkel:    Lehrbuch  der  Petrographie.     Leipzig,  2  Aufl.,  1893, 1,  107. 


558  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  500 

needle  is  held  for  a  moment  in  a  vial  of  benzol.  This  dissolves  the  balsam 
and  permits  the  mineral  fragments  to  accumulate  in  the  bottom  of  the  flask. 
Instead  of  Canada  balsam,  the  needle  may  be  dipped  in  glycerine  and  the 
minerals  transferred  to  water. 

500.  Separation  by  Chemical  Means. — Ordinarily  it  is  not  necessary  to 
make  chemical  separations  of  mineral  constituents.  At  most  it  may  be 
necessary  to  remove  carbonates,  which  is  readily  done  by  means  of  dilute 
hydrochloric  acid.  Separation  methods  will  readily  suggest  themselves  to 
a  chemist,  to  others  the  directions  which  could  be  given  within  the  limits 
of  this  book  would  be  useless. 


CHAPTER  XL 
MICROCHEMICAL  REACTIONS 

501.  General  Microchemical  Reactions. — For  petrogra  phical  purposes 
general  microchemical  reactions  for  the  determination  of  the  elements  are 
used  but  little.     The  subject  is  a  study  in  itself,  and  the  student  is  referred 
especially  to  Boricky's  paper  and  Behrens'  English  translation  of  his  own 
"Manual  of  Microchemical  Analysis,"  published  in  1894.     For  convenience, 
a  bibliography  is  given  at  the  end  of  this  chapter. 

CHEMICAL  REACTIONS  ON  ROCK  SLICES 

502.  Apparatus. — Certain  special  materials  and  apparatus  are  required  in 
microchemical  researches,  those  necessary  for  the  examination  of  reactions 
on  thin  rock  sections  are: 

Microscope. — Special  chemical  microscopes  have  been  designed,  some 
of  them  inverted.  A  cheap  microscope  is  all  that  is  necessary,  provided 
it  is  fitted  with  nicol  prisms.  No  higher  magnification  than  200  diameters 
is  required.  If  the  objective  used  is  corrected  for  cover-glasses,  it  will 
add  to  the  clearness  of  the  image  if  an  ordinary  cover-glass  is  stuck  to  the 
lower  lens  with  cedar  oil  or  glycerine.  This  serves  to  counteract  the  effect 
of  the  removal  of  the  cover-glass  from  the  rock  section,  and  also  acts  as  a 
protecting  screen  for  both  the  objective  and  its  casing  against  the  reagents 
used.  Another  method  is  to  work  under  a  kind  of  table  made  of  a  large 
cover-glass  resting  on  pieces  of  cork  attached  at  the  corners.  For  most 
observations,  objectives  with  a  clear  working  distance  of  3  cm.  should  be 
used. 

Canada  Balsam. — For  immediate  use,  it  is  sometimes  desirable  to  have 
Canada  balsam  dissolved  in  some  medium  which  will  evaporate  quickly, 
without  heat,  leaving  the  balsam  hard.  It  may  be  prepared  by  heating 
the  balsam  in  a  shallow  dish  until  a  sample,  cooled  in  water,  is  of  sufficient 
hardness.  It  should  then  be  broken  up  and  dissolved  in  bisulphide  of 
carbon  or  ether  to  the  consistency  of  cream.  The  first  has  an  unpleasant 
odor,  and  the  latter,  if  dried  in  a  damp  atmosphere,  may  become  turbid, 
owing  to  the  absorption  of  moisture;  a  mishap  not  likely  to  happen  in  our 
steam-heated  laboratories,  however.  If  the  balsam  thus  prepared  is  too 
hard,  more  or  less  soft  balsam  may  be  added  to  it.  Fifteen  minutes  should 
be  sufficient  for  the  material  to  harden. 

559 


560  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  503 

Capillary  Tubes  and  Pipettes. — For  transferring  reagents  to  slides  or 
removing  worked  out  liquids,  capillary  tubes  and  pipettes  made  of  glass 
are  useful.  Being  readily  made,  they  may  be  thrown  away  after  using, 
thus  doing  away  with  cleaning. 

Glass  Dropper. — A  glass  rod  drawn  out  rather  thin  and  with  a  slightly 
enlarged  end  is  useful  in  taking  large  drops  from  a  reagent  bottle. 

Platinum  Dropper. — A  most  convenient  dropper  is  made  of  a  platinum 
wire,  0.5  mm.  in  diameter  and  bent  into  a  small  hook  or  loop  at  the  end. 
If  quickly  withdrawn  from  a  reagent  bottle,  a  large  drop  will  be  carried 
away;  if  slowly,  a  small  one,  provided  the  wire  is  clean.  Thicker  straight 
wires  may  be  used  instead,  and  have  the  advantage  of  being  cleaned  more 
readily. 

Burner. — A  small  Bunsen  burner  giving  a  flame  5  to  10  mm.  in  length  is 
useful. 

Water-bath. — A  small  water-bath,  about  the  size  of  a  cigar  box,  is  very 
convenient.  One  may  be  improvised  from  a  large  evaporating  dish  covered 
with  a  glass  plate  over  which  a  small  pasteboard  box  is  inverted.  The 
slide  may  be  laid  directly  upon  the  glass  plate,  upon  a  piece  of  pasteboard 
within  the  box,  or  upon  the  box  itself,  depending  upon  the  degree  of  heat 
desired. 

503.  Preparing  the  Slide. — The  chemical  determination  of  certain  prop- 
erties of  minerals  is  often  necessary,  and  special  processes  and  methods 
must  be  used.  Ordinarily  these  examinations  must  be  made  on  rock  sections 
which  are  also  to  be  used  for  the  general  determinations  of  the  rock,  and  it 
is  usually  desirable  that  the  section  be  spoiled  as  little  as  possible  by  the 
operation. 

To  make  a  chemical  test  on  a  mineral  it  is  necessary,  of  course,  that 
the  cover-glass  be  removed  from  that  portion  of  the  slide.  This  can  be  done 
roughly  by  cutting  on  the  cover-glass,  with  a  marking  diamond,  or  more 
neatly,  with  a  slide  marker  (Figs.  760-761),  a  circle  around  the  mineral, 
placing  the  slide  for  a  moment,  cover-glass  downward,  on  a  heated  plate 
(Fig.  755)  or  a  water-bath,  and  lifting  the  small  circular  section  by  means  of 
a  needle.  The  balsam  underlying  the  opening  is  removed  by  placing  a  few 
drops  of  alcohol  upon  it  by  means  of  a  camel-hair  brush,  allowing  it  to  act 
for  a  short  time,  removing  the  white  gum  with  a  rolled-up  piece  of  filter 
paper,  applying  more  alcohol,  and  so  on  until  the  mineral  is  uncovered. 

If  the  mineral  grain  to  be  treated  is  very  small  and  a  smaller  opening  is 
desired,  the  old  cover-glass  must  first  be  removed  by  placing  the  slide  with 
the  cover-glass  downward  on  the  hot  plate  until  the  balsam  is  softened,  then 
sliding,  not  lifting,  it  from  the  preparation.  If  a  few  drops  of  turpentine 
are  placed  on  the  upper  side  of  the  thin  section,  its  evaporation  will  assist 
in  keeping  the  balsam  film  between  rock  slice  and  mounting  slip  hard.  Care 


ART.  504]  MICROCHEMICAL  REACTIONS  561 

must  be  used  not  to  heat  the  slide  too  much,  otherwise  it  is  likely  to  go  to 
pieces  on  the  mount.  A  few  drops  of  well-cooked  balsam  are  now  placed 
over  the  section,  allowed  to  spread,  and  then  to  cool.  In  a  new  cover-glass, 
a  hole  is  drilled,  or  bitten  by  acid  according  to  a  method  to  be  explained, 
in  such  a  position  that  when  it  lies  over  the  mineral  section  the  cover-glass 
will  cover  approximately  all  of  the  rock  slice.  This  glass  is  now  placed  over 
the  rock  slice  and  shoved  about,  under  the  microscope,  until  the  hole  lies 
over  the  mineral  to  be  examined,  when  slide  and  cover  are  carefully  removed 
and  heated  until  the  balsam  softens  enough  to  begin  to  push  through  the 
hole.  After  cooling,  the  balsam  is  removed  from  above  the  mineral  by  the 
method  previously  described. 

Holes  may  be  made  in  cover-glasses  by  means  of  a  diamond  drill.  Since 
such  is  usually  not  at  hand  in  the  laboratory,  a  different  process  is  necessary. 
The  cover-glass  may  be  dipped  in  melted  wax1  and,  after  cooling,  by  means  of  a 
needle  point,  a  circle  of  the  desired  size  (1/4  to  3/4  mm.)  may  be  scratched 
upon  it.  A  few  drops  of  hydrofluoric  acid,  renewed  as  often  as  necessary,  will 
soon  bite  through  the  glass,  leaving  a  cone-shaped  hole.  If  the  small  opening 
is  not  quite  large  or  round  enough,  it  may  be  enlarged  by  means  of  a  needle. 
The  wax  may  be  removed  from  the  remainder  of  the  slide  by  means  of 
hot  water.  A  number  of  such  cover-glasses,  with  holes  in  various  parts, 
may  be  prepared  beforehand  and  kept  in  stock.  When  placed  over  a  slide 
the  smaller  end  of  the  funnel-shaped  hole  should  lie  against  the  rock  slice, 
the  larger  side  up.  The  remainder  of  the  slide,  being  protected  by  the  glass 
and  the  Canada  balsam,  will  not  be  acted  upon  by  the  reagents  used.  If 
hydrofluoric  acid  is  the  reagent,  a  piece  of  thin,  perforated  platinum  foil 
should  be  substituted  for  the  glass,  its  proper  position  on  the  slide  being 
readily  determined,  under  the  microscope,  by  first  centering  the  mineral 
under  the  cross-hairs  and  then  sliding  the  platinum  foil  into  place. 

Another  method2  for  protecting  the  remainder  of  the  slice  is  to  cover  it, 
after  removing  the  cover-glass,  with  rather  a  thick  coating  of  balsam  dissolved 
in  ether.  In  a  few  hours  the  balsam  will  be  hard,  and  a  hole  may  be  scratched 
through  it  directly  over  the  mineral  to  be  examined. 

504.  Microchemical  Filtrations. — The  most  satisfactory  device  for 
fitering  small  quantities  of  liquid  is  that  of  Streng.3  The  slide,  with  the 
liquid  to  be  filtered  upon  it,  is  placed  on  a  small  box  turned  upside  down  and 

1  A.  Streng:  Ueber  eine  Methode  zur  Isolirung  der  Miner  alien  eines  DunnsMifs  behufs 
ihrcr  mikroskopisch-chemischen    Untersuchung.     Ber.   oberhess.      Gesell.    Giessen,   XXII 
(1883),  260-262. 

Idem:  Ueber  einige  mikroskopisch-chemische  Reaktionen.     Neues  Jahrb.,  1885  (I),  26. 
Idem:  Erundenmg.     Neues  Jahrb.,  1885  (I),  174-175. 

2  Arthur  Wichmann:  Ueber  eine  Methode  zur  Isolirung  ion  Miner  alien  behufs  ihrer 
mikrochemischen  Untersuchung.     Zeitschr.  f.  wiss.  Mikroskop.,  I  (1884),  417-419. 

A.  Streng:  Erunderung.     Neues  Jahrb.,  1885  (I),  174-175. 

3  A.  Streng:  Anleitung  zur  Bestimmung  der  Miner  alien,  65.* 

36 


562  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  505 

slightly  inclined.  With  a  width  of  5  cm.  the  box  should  be  about  10  mm. 
high  on  one  side  and  12  mm.  on  the  other.  A  piece  of  filter  paper,  cut  in 
the  form  of  a  letter  Y,  with  a  width  of  from  i  to  2  mm.  and  a  length  of  10 
to  25  mm.,  is  so  placed  that  the  reentrant  angle  of  the  forked  end  is  in 
contact  with  the  liquid  to  be  filtered  while  the  lower  end  touches  a  clean  slide 
placed  near  the  box.  To  retain  it  in  place,  the  upper  arms  may  first  be 
slightly  moistened.  The  liquid  will  now  run  through,  perfectly  clear  and 
transparent,  while  the  solids  remain  above.  For  the  filtration  to  be  successful 
there  must  be  enough  fluid  so  that  it  is  not  all  retained  in  the  filter,  at  least 
o.oi  c.c.  being  necessary,  if  it  is  not  to  be  diluted,  with  paper  of  the  size 
mentioned.  The  solid  portion  may  be  washed  by  the  gradual  addition  of 
water,  the  lower  end  of  the  filter  strip  being  placed 
on  a  folded  piece  of  filter  paper  to  absorb  the  wash 
water. 

For  larger  quantities  of  liquid  the  Haushofer1 
filter  is  the  best.  It  consists  of  two  thick  glass 
tubes  (a  and  b,  Fig.  745)  with  an  inner  diameter  of 
about  4  mm.  The  abutting  ends  c  are  smoothly 
ground  and  are  kept  in  contact  by  the  screw  S. 
Between  the  two  ends  c  is  placed  a  double  piece  of 
FIG.  745.-Haushofer  filter.  mtej.  psiper  w[ih  &  diameter  a  couple  of  millimeters 

greater  than  that  of  the  outside  of  the  tube,  and 

clamped  in  place  by  the  screw  S.  A  rubber  tube  is  attached  to  e,  and  the 
liquid  to  be  filtered  is  poured  into  the  funnel-shaped  end  of  the  upper  tube. 
By  sucking  through  the  rubber  tube,  the  liquid  is  rapidly  filtered  into  b. 
The  filtrate  is  removed  by  opening  the  stopper  d,  the  precipitate  remain- 
ing in  a  compact  ring  4  mm.  in  diameter  on  the  filter  paper. 

505.  Gelatinizing  and  Staining  Minerals. — A  general  gelatinization  test 
may  be  made  by  entirely  removing  the  cover-glass  from  the  slide  and  cleaning 
off  the  Canada  balsam  by  means  of  alcohol  or  ether.  Usually  it  will  be  found 
sufficient  if  only  a  part  of  the  slide  is  uncovered,  which  may  be  done  by  cutting 
across  with  a  diamond,  heating,  and  sliding  the  cover-glass  from  one  side. 
If  the  cover-glass  is  entirely  removed,  the  reagent  may  be  confined  to  a 
portion  of  the  slide  by  surrounding  it  with  a  ring  of  hardened  balsam. 

A  few  drops  of  hydrochloric  acid  are  now  spread  in  a  thin  film  over  the 
part  of  the  slide  to  be  tested.  Only  enough  acid  is  used  to  make  a  surface 
etching,  otherwise  the  resulting  gelatine  will  spread  over  the  surrounding 
unattacked  minerals  and  cause  confusion.  It  is  better  to  make  several 
successive  trials  than  to  cause  too  vigorous  action  at  once.  After  allowing 
the  acid  to  act  for  a  short  time,  perhaps  with  gentle  heating,  it  is  washed  off, 

1  K.  Haushofer:  Beitrage  zur  mikroskopisch-chemischen  Analyse.  Sitzb.  Akad.  Wiss. 
Miinchen,  XV  (1885),  224-226. 


ART.  506]  MICROCHEMICAL  REACTIONS  563 

care  being  taken  not  to  remove  the  gelatine  coating.  It  may  be  well  to  use 
a  few  drops  of  dilute  ammonia  to  neutralize  the  acid. 

The  section  will  now  be  found  to  have  a  thin  film  of  gelatine  over  such 
minerals  as  gelatinize  with  acid.  To  make  it  more  visible  it  is  necessary  to 
stain  it.  This  is  done  by  covering  the  slide  with  an  aqueous  solution  of  some 
dye  and  allowing  it  to  act  for  about  15  minutes.  Sometimes  slight  heating 
will  aid  the  staining,  as  will  also  a  trace  of  ammonia.  The  slide  is  washed 
to  remove  the  stain  from  such  minerals  as  were  not  attacked  by  the  acid, 
and  it  is  examined  under  the  microscope.  Gelatinized  minerals  will  have 
taken  the  stain,  as  will  also  cracks  in  other  minerals.  If  it  is  found  that  the 
action  was  not  continued  long  enough,  it  is  repeated,  the  new  acid  destroying 
the  dye. 

The  coloring  matter  generally  used  is  fuchsine,  first  recommended  by 
Behrens1  and  afterward  by  Haushofer.2  While  it  has  great  staining  power, 
it  fades  in  the  light  and  is  not  permanent  in  the  presence  of  Canada  balsam. 
Malachite  green  surpasses  fuchsine  in  staining  powrer,  and  is  permanent. 
Methylene  blue  is  nearly  equal  in  staining  power  but  is  likely  to  form  films 
on  rough  surfaces. 

If  the  solvent  is  to  be  examined,  the  acid  is  allowed  to  act  for  a  longer 
time  and  is  then  removed  with  a  capillary  tube,  placed  on  an  object  glass, 
and  tested  for  various  chemical  reactions.3 

SPECIAL  REACTIONS,  CHIEFLY  ON  THIN  SECTIONS 

506.  Hauynite,  Noselite,  Sodalite,  Melilite,  and  Zeolites. — Minerals  of 
the  sodalite-noselite-hauynite  group  may  be  treated  with  acid  and  the  solvent 
removed  by  means  of  a  capillary  tube  and  placed  upon  a  clean  object  glass. 
Sodalite  will  be  found  to  dissolve  without  gelatinization  in  HNO3,  and  cubes 
of  NaCl  form  upon  drying  the  solution.4  If  a  few  drops  of  a  dilute,  slightly 
acid  solution  of  lead  acetate  are  placed  upon  a  thin  section  of  sodalite  upon 
which,  previously,  a  few  drops  of  dilute  Cl-free  HNO3  or  acetic  acid  have 
been  placed,  thin,  flat  needles  of  strongly  refracting  lead  chloride  form  over 
it.5  Noselite  and  hauynite  gelatinize  in  thin  sections  with  HC1.  Upon 

1  H.   Behrens:  Mikroskopische    Untersuchungen  uber  die  Opale.     Sitzb.  Akad.  Wiss. 
Wien,  LXIV,  I  Abth.  (1871),  521. 

2  K.  Haushofer:  Mikroskopische  Reactionen.     Eine  Anleitung  zur  Erkennung  -oerschied- 
ener  Elemente  unter  dem  Mikroskop  als  Supplement  der  qualitativen  Analyse.  Munchen,  1885.* 

3  H.  Behrens:  A   manual  of  microchemical  analysis.     London,   1894.     See  also  other 
papers  mentioned  in  the  preceding  pages  and  in  the  General  Bibliography  at  the  end  of 
the  chapter. 

4  G.   A.   Sauer:  Untersuchungen  uber  phonolithische  Gesteine  der  Kanarischen  Inseln. 
Zeitschr.  f.  d.  gesammten  Xaturw.,  XIII  (1876),  322.* 

5  Franz    F.    Graeff :    Miner alogisch-petrographische    Unter suchung   von  Eldolithsyeniten 
von  Serra  de  Tingud,  Provinz  Rio  de  Janeiro,  Brasilien.     Neues  Jahrb.,  1887  (II),  230. 

G.  Freda:  Sulle  masse  trachitiche  rinvenute  nei  recenti  trafori  delle  colline  di  Napoli. 
Rendiconti  della  Acad.  di.  Xapoli,  III  (1889),  (2),  39.* 


564  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  507 

drying  the  solution  derived  from  the  former,  much  NaCl  (cubes)  and  a  little 
CaSCU  (needles)  separates;  from  hauynite  much  CaSCU  separates.1  Sodalite 
and  noselite  may  be  separated  by  placing  over  them  a  drop  of  dilute  acetic 
acid  (one  part  acid  to  three  or  four  of  water)  to  which  a  little  BaCl2  solution 
has  been  added.  To  prevent  complete  drying,  the  section  and  a  watch 
crystal  containing  some  of  this  fluid  are  set  away  under  a  bell  jar.  It  will 
be  found,  according  to  Osann,2  that  the  sodalite  will  remain  clear,  although 
it  will  be  etched,  while  the  noselite  will  be  covered  with  an  opaque  film  of 
BaSO4. 

That  colorless  members  of  the  hauynite  groups  may  be  colored  blue  by 
heating,  was  shown  by  Vogelsang.3  The  same  result  was  obtained  by  Knop4 
by  heating  the  uncovered  thin  section  in  a  closed  vessel,  in  the  bottom  of  which 
was  placed  a  pinch  of  flowers  of  sulphur. 

Analcite  gelatinizes  with  HNO3.  It  differs  from  sodalite  in  that  no 
cubical  crystals  of  sodium  chloride  form  from  the  evaporated  solution. 

Noselite,  hauynite,  and  analcite  will  take  stain  readily  since  they  gelatinize 
easily.5 

Melilite  gelatinzes  readily  with  HC1.  If  a  drop  of  H2SO4  be  added  to 
the  hydrochloric  acid  solution,  crystals  of  gypsum  are  formed  on  the  slide.6 

507.  Nephelite,  Cancrinite,  and  Hydronephelite. — If  a  thin  section  con- 
taining nephelite  and  cancrinite  is  heated,  no  changes  appear  in  the  former 
but  the  latter  becomes  cloudy,  probably  due  to  the  driving  off  of  the  CO2.7 
Nephelite  gelatinizes  readily  with  HC1  and  takes  stain.  From  the  solution, 
cubes  of  NaCl  are  formed.  Cancrinite  gelatinizes  after  it  is  acted  upon  by 
warm  HC1.  There  is  a  slight  evolution  of  CO2  which  may  readily  be  observed 
under  a  cover-glass  as  described  under  carbonates  (Art.  510). 

1  G.  A.  Sauer.    Op.  cit. 

2  A.  Osann:    Ueber  ein  Mineral  der  Nosean-Hauyn-Gruppe  im  Eldolithsyenit  von  Mon- 
treat.     Neues  Jahrb.,  1892  (I),  224. 

3  H.  Vogelsang:    Ueber  die  natiirlichen   Ultramarineverbindungen.     Versl.  en  Meded. 
Akad.  Weten.  Amsterdam,  VII  (1873),  161-199. 

4  A.   Knop:    Ueber  eine  mikrochemische  Reaction   auf  die  Glieder  der  Hauynfamilie. 
Neues  Jahrb.,  1875,  74-76. 

6  J.  Lemberg:  Zur  mikrochemischen  Untersuchung  einiger  Miner  ale.  Zeitschr.  d. 
deutsch.  geol.  Gesell.,  XLII  (1890),  738-740. 

H.  Dressel:  Mittheilungen  torn  Laacher  See.     Neues  Jahrb.,  1870,  565. 

G.  vom  Rath:  Miner alogisch-geognostische  Fragmente  aus  Italien.  Zeitschr.  d.  deutsch. 
geol.  Gesell.,  1866,  547. 

6  Alfred  Stelzner:  Ueber  Melilith  und  Melilithbasalte.  Neues  Jahrb.,  B.  B.,  II  (1883),  382. 

7  A.  E.  Tornebohm:    Om  den  s.  k.  Fonolitenfraan  Elfdalen,  dess  klyftort  och  fb'rekomstsatt. 
Geol.  Foren.  i  Stockholm  Forh.,  VI  (1883),  383-405. 

E.  Cohen:  Review  of  above.     Neues  Jahrb.,  1883  (II),  370-371. 

A.  Streng:  Ueber  die  mikroskopische  Unterscheidung  von  Nephelin  und  Apatit.  T.  M. 
P.  M.,-  1876,  168-169. 

Idem:  Ueber   einige   mikroskopisch-chemische   Reaktionen.     Neues    Jahrb.,    1885    (I), 

2Q-33- 


ART.  510]  MICROCHEMICAL  REACTIONS  565 

Hydronephelite  is  soluble  in  HC1,  and,    upon  evaporation,    gelatine  is 
formed. 

In  making  the  gelatinization  test,  care  should  be  taken  not  to  mistake 
the  mineral  from  which  the  gelatine  was  derived. 

508.  Olivine  Family. — Olivine  gelatinizes  slowly  with  cold  and  rapidly 
with  hot  HC1  or  H2SO4.     The  iron  rich  members  are  more  readily  acted  upon 
than  are  the  iron  poor. 

509.  Apatite. — Apatite  is  easily  soluble  in  HC1  or  HNO3.     If  ammonium 
molybdate  is  added  to  the  solution,  a  yellow  precipitate,  consisting  of  iso- 
metric  crystals,  is   formed.1     If  dilute  H2SO4  is  added  to  the  nitric  acid 
solution,  gypsum  crystals  develop  upon  evaporation. 

510.  Carbonates. — Upon  the  addition  of  acids  to  carbonates,  an  effer- 
vescence arises  from  the  escape  of  the  carbon  dioxide.     This  breaking  up 
occurs  in  some  carbonates  upon  the  addition  of  acetic  acid,  in  other  with 
cold  hydrochloric  acid,  and  in  still  others  only  with  hot.     If  the  amount  of 
carbonate  is  small,  the  escape  of  the  gas  may  not  be  noticed.     In  such  cases 
a  drop  of  water  may  be  placed  on  the  section  and  over  it  a  cover-glass.     If 
a  drop  of  acid  is  brought  to  the  edge  of  the  latter,  it  will  gradually  diffuse 
through  the  water.     The  cover-glass  will  prevent  the  escape  of  the  gas  and, 
if  the  latter  is  in  small  amount,  will  confine  it  immediately  above  the  mineral 
from  which  it  is  being  evolved,  thus  permitting  its  study  under  the  micro- 
scope.    The  solvent  may  be  removed  writh  a  capillary  tube  and  studied,  if 
desired. 

Separation  of  Calcite,  Dolomite,  and  Magnesite. — Calcite  is  acted  upon  by 
acetic  or  hydrochloric  acid  even  when  they  are  cold.  Dolomite  and  magnesite 
require  hot  hydrochloric  acid. 

Calcite  and  dolomite  may  be  separated  by  Lemberg's2  method  which 
depends  upon  the  fact  that  aluminium  hydroxide  is  quickly  and  completely 
precipitated  from  solutions  of  aluminium  salts  by  calcite  and  very  slowly 
by  dolomite.  Further,  if  the  precipitation  takes  place  in  the  presence  of 
coloring  matter,  it  generally  combines  with  it  to  form  an  insoluble  coating. 
To  60  parts  of  water  are  added  4  parts  of  dry  aluminium  chloride  and  6 
parts  of  logwood  (haematoxylon  campechianum).  The  ingredients  are  boiled 
together  for  twenty-five  minutes  with  stirring,  the  evaporated  water  being 
constantly  replaced.  When  cold,  the  deep  violet  solution  is  filtered.  If  a 
few  drops  of  this  solution  are  placed  on  a  thin  section  of  calcite,  are  allowed 
to  stand  from  five  to  ten  minutes,  and  are  then  carefully  washed  off  with 
water,  the  section  will  be  colored  violet.  A  section  of  dolomite  with  the 

1  A.  Streng:  Op.  cit.,  168. 

A.  Stelzner:  Ueber  Mclilith  und  Melilithbasalte.     Neues  Jahrb.,  B.  B.,  II  (1883),  382. 

2  J.  Lemberg:    Zur  mikroskopischen  Unter suchung  von  Calcit,   Dolomit   und  Predazzit. 
Zeitschr.  d.  deutsch.  geol.  Gesell.,  XC  (1888),  357-359. 


566  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  510 

same  treatment  remains  unchanged,  and  only  after  twenty  minutes'  treatment 
does  it  show  faint  stains  in  spots. 

An  earlier  method  of  Lemberg1  consisted  in  treating  both  minerals  with 
ferric  chloride.  A  solution  of  one  part  crystallized  hydrochloric-acid-free 
ferric  chloride  (Fe2Cl6+i2H2O)  in  ten  parts  of  water  is  used.  If  any 
basic  salt  separates  the  solution  is  filtered.  When  this  solution  is  placed  on 
calcite,  the  latter,  within  a  minute,  precipitates  the  iron  as  a  hydroxide. 
If  the  slide  or  mineral  grains  are  washed,  this  precipitate  appears  as  a  brown 
coating  which  becomes  black,  by  changing  to  FeS,  if  a  solution  of  ammonium 
sulphide  ((NH^S)  is  poured  over  it.  Dolomite  treated  with  ferric  chloride 
for  the  same  length  of  time  shows  no  change  to  the  eye  although  pouring 
ammonium  sulphide  over  it  changes  it  to  pale  green  by  incident  light  while 
it  remains  colorless  by  transmitted.  Brucite  acts  like  dolomite. 

Sections  treated  by  the  method  just  described  do  not  show  a  permanent 
coloration,  since  the  FeS  readily  oxidizes.  It  may,  however,  be  made  per- 
manent as  follows :  Immediately  after  the  ammonium  sulphide  has  converted 
the  FeOH  to  FeS,  it  is  washed  off  the  slide  and  a  concentrated  solution  of  potas- 
sium ferricyanide  is  quickly  poured  over  it  and  allowed  to  remain  about 
half  a  minute.  It  is  then  renewed  and  allowed  to  remain  eight  minutes. 
The  resulting  Prussian  blue  is  permanent.  If  time  is  allowed  to  elapse  before 
the  addition  of  the  potassium  ferricyanide,  oxidation  will  set  in  and  spoil  the 
reaction. 

,  Linck2  prepared  a  solution  of  20  c.c.  of  ammonium  phosphate  in  30  c.c. 
of  dilute  acetic  acid.  If  this  preparation  is  allowed  to  remain  on  a  slide  of 
pure  calcite,  complete  solution  will  take  place,  while  slides  of  dolomite  or 
magnesite  are  but  slightly  altered  on  the  surface,  being  immediately  protected 
from  further  action  of  the  acid  by  a  coating  of  magnesium  ammonium 
phosphate.  The  film  forms  with  as  little  as  10  to  15  per  cent,  of  MgCO3. 
The  solution  should  be  allowed  to  act  for  twenty-four  hours. 

Another  reaction  depending  upon  the  precipitation  of  iron  hydroxide  or 
copper  carbonate,  was  given  first  by  Lemberg3  and  later  by  Hinden.4  If 
i  grin,  of  powdered  calcite  is  thoroughly  shaken  up  with  5  c.c.  of  a  10  per 
cent,  solution  of  iron  chloride,  a  violent  effervescence  takes  place,  and  the 
solution  becomes  dark  reddish  brown.  After  two  or  three  minutes  the  solu- 
tion in  the  test-tube  becomes  thick  and  jelly-like  and  of  a  rust-brown  color, 
due  to  the  separation  of  FeOH.  If  5  c.c.  of  a  5  per  cent,  solution  of  potassium 
thiocyanate  (KCNS)  be  now  added  to  the  solution,  no  further  change  takes 

1  J.  Lemberg:    Zur  mikrochemischen  Untersuchung  von  Calcit,  Dolomit  und  Predazzit. 
Zeitschr.  d.  deutsch.  geol.  Gesell,  XXXIX  (1887),  489-492. 

2  G.  Linck:    Geognostisch-petrographische  Beschreibung  des  Grauwackengebiets  von  Weiler 
bei  Weissenburg.    Abh.  zur  geol.  Spezialkarte  von  Elsass-Lothringen.,  Ill  (1884),  17.* 

3  J.  Lemberg:    Op.  cit.,  Zeitschr.  d.  deutsch.  geol.  Gesell.,  XXXIX  (1887),  489-492. 

4  Fritz  Hinden:    Neue  Reaktionen  zur  Under  scheidung  von  Calcit  und  Dolomit.     Ver- 
handl.  d.  Naturforsch.  Gesell.  in  Basel,  XV  (1903),  Hft.  2. 


ART.  510]  MICROCHEMICAL  REACTIONS  567 

place,  since  all  of  the  iron  was  previously  precipitated  (i  grin,  of  calcite 
will  precipitate  the  iron  from  14  c.c.  of  a  10  per  cent,  ferric  chloride  solution.) 

If,  in  the  same  manner,  ferric  chloride  is  added  to  dolomite  powder,  no 
change  takes  place  unless  the  solution  is  heated.  If  5  c.c.  of  the  potassium 
thiocyanate  solution  are  added  to  the  solution,  not  previously  heated,  the  well- 
known  deep-red  iron  reaction  color  appears.  This  test  may  be  used  quanti- 
tatively. To  i  grm.  of  the  rock  powder,  in  a  flask,  there  is  added  5  c.c. 
of  a  5  per  cent,  potassium  thiocyanate  solution  and  then  enough  ferric  chloride 
from  a  burette  to  give  a  permanent  blood-red  color,  the  ferric  chloride  being 
added  a  little  at  a  time  with  vigorous  shaking.  By  experiment  it  was 
found  that  i  c.c.  of  the  ferric  chloride  solution  represented  8  per  cent.  CaCOs 
in  the  mineral  examined. 

The  Hinden  test  may  be  made  directly  on  a  hand  specimen  or  thin 
section,  the  ferric  chloride  giving,  after  one  or  two  minutes,  a  dark  red-brown 
color  to  calcite  while  dolomite  shows  no  change.  Magnesium  rich  calcite 
shows  a  more  or  less""pale  brown  color,  depending  upon  the  amount  of-  the 
calcium  carbonate  present. 

A  similar  reaction  takes  place  by  boiling  i  grm.  of  calcium  carbonate  or 
dolomite  with  5  c.c.  of  a  10  per  cent,  solution  of  CuSO4.  The  former  gives 
the  blue  color  of  basic  copper  carbonate  while  the  latter  shows  no  change. 
Ammonia  added  to  the  filtered  or  decanted  solution  derived  from  the  calcite 
shows  no  change,  while  that  from  the  dolomite  becomes  dark  brown. 

If  any  hydroxide  of  iron  was  present  in  the  slide  itself,  this  acts  as  a 
disturbing  cause,  especially  if  through  the  addition  of  (NH^S  it  is  changed 
to  FeS.  To  overcome  this,  and  likewise  to  make  permanent  mounts, 
Lemberg  transformed  the  .FeS  into  Turnbull's  blue  (Fe3[Fe(CN)*6]2)  by 
treating  it  with  potassium  ferricyanide  (K3Fe(CN)6). 

Both  Lemberg's  and  Link's  methods  are  practically  useless  for  rocks 
in  which  the  carbonate  is  very  finely  distributed,  the  stain  not  holding  with 
short  action  and  sinking  into  cracks  with  longer  action.  Heger,1  therefore, 
proposed  a  method  which  consists  in  treating  the  section  with  dilute  HC1 

(2-3  c.c.  of  — )  to  which  a  few  drops  of  potassium  ferricyanide  has  been 

added.  The  reaction  should  be  watched  under  the  microscope.  If  calcite 
is  present  the  reaction  is  great  enough  to  cause  effervescence  and  the  acid 
should  be  washed  off  after  a  few  seconds.  The  calcite  will  be  found  colored 
a  deep  blue  if  it  is  not  entirely  free  from  iron  as  an  impurity.  The  slides  should 
then  be  washed  gently  in  water.  With  dolomite  or  other  carbonates  the 
reaction  is  much  slower. 

Separation  of  Calcite  from  Hydromagnesite  and  Brucite. — If  grains  of  calcite, 
hydromagnesite,  and  brucite  are  heated  until  the  latter  two  lose  their 

1 W.  Heeger:  Ueber  die  mikrochemische  Untersuchung  fein  verteilter  Carbonate  im 
Gesteinsschlijf.  Centralbl.  f.  Min.,  etc.,  1913,  44-51. 


558  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  500 

needle  is  held  for  a  moment  in  a  vial  of  benzol.  This  dissolves  the  balsam 
and  permits  the  mineral  fragments  to  accumulate  in  the  bottom  of  the  flask. 
Instead  of  Canada  balsam,  the  needle  may  be  dipped  in  glycerine  and  the 
minerals  transferred  to  water. 

500.  Separation  by  Chemical  Means. — Ordinarily  it  is  not  necessary  to 
make  chemical  separations  of  mineral  constituents.  At  most  it  may  be 
necessary  to  remove  carbonates,  which  is  readily  done  by  means  of  dilute 
hydrochloric  acid.  Separation  methods  will  readily  suggest  themselves  to 
a  chemist,  to  others  the  directions  which  could  be  given  within  the  limits 
of  this  book  would  be  useless. 


CHAPTER  XL 
MICROCHEMICAL  REACTIONS 

501.  General  Microchemical  Reactions. — For  petrogra  phical  purposes 
general  microchemical  reactions  for  the  determination  of  the  elements  are 
used  but  little.     The  subject  is  a  study  in  itself,  and  the  student  is  referred 
especially  to  Boricky's  paper  and  Behrens'  English  translation  of  his  own 
"Manual  of  Microchemical  Analysis,"  published  in  1894.     For  convenience, 
a  bibliography  is  given  at  the  end  of  this  chapter. 

CHEMICAL  REACTIONS  ON  ROCK  SLICES 

502.  Apparatus. — Certain  special  materials  and  apparatus  are  required  in 
microchemical  researches,  those  necessary  for  the  examination  of  reactions 
on  thin  rock  sections  are: 

Microscope. — Special  chemical  microscopes  have  been  designed,  some 
of  them  inverted.  A  cheap  microscope  is  all  that  is  necessary,  provided 
it  is  fitted  with  nicol  prisms.  No  higher  magnification  than  200  diameters 
is  required.  If  the  objective  used  is  corrected  for  cover-glasses,  it  will 
add  to  the  clearness  of  the  image  if  an  ordinary  cover-glass  is  stuck  to  the 
lower  lens  with  cedar  oil  or  glycerine.  This  serves  to  counteract  the  effect 
of  the  removal  of  the  cover-glass  from  the  rock  section,  and  also  acts  as  a 
protecting  screen  for  both  the  objective  and  its  casing  against  the  reagents 
used.  Another  method  is  to  work  under  a  kind  of  table  made  of  a  large 
cover-glass  resting  on  pieces  of  cork  attached  at  the  corners.  For  most 
observations,  objectives  with  a  clear  working  distance  of  3  cm.  should  be 
used. 

Canada  Balsam. — For  immediate  use,  it  is  sometimes  desirable  to  have 
Canada  balsam  dissolved  in  some  medium  which  will  evaporate  quickly, 
without  heat,  leaving  the  balsam  hard.  It  may  be  prepared  by  heating 
the  balsam  in  a  shallow  dish  until  a  sample,  cooled  in  water,  is  of  sufficient 
hardness.  It  should  then  be  broken  up  and  dissolved  in  bisulphide  of 
carbon  or  ether  to  the  consistency  of  cream.  The  first  has  an  unpleasant 
odor,  and  the  latter,  if  dried  in  a  damp  atmosphere,  may  become  turbid, 
owing  to  the  absorption  of  moisture;  a  mishap  not  likely  to  happen  in  our 
steam-heated  laboratories,  however.  If  the  balsam  thus  prepared  is  too 
hard,  more  or  less  soft  balsam  may  be  added  to  it.  Fifteen  minutes  should 
be  sufficient  for  the  material  to  harden. 

559 


570  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  511 

1885.  H.     Behrens:   Sur     I' analyse    microchimique   des    miner  aux.     Ann.     Ecole    Polyt. 

Delft.     Leiden,  I  (1885),  176-212.* 
1885.  A.    Streng:  Mikroskopisch-chemische    Bestimmung   von    Kobalt    und    Nickel.     Ber. 

oberhess.  Gesell.,  Giessen,  XXIV  (1885),  58-59. 
1885.  Idem:  Ueber  eine  neue   mikroskopisch-chemische   Reaction   auf  Natrium.     Ibidem, 

56-58. 
1885.  Idem:  Ueber  einige  mikroskopisch-chemische  Reactionen.     Ibidem,  54-55. 

1885.  Idem:  Ueber  einige  mikroskopisch-chemische  Reactionen.     Neues  Jahrb.,   1885   (I)> 

21-42. 

1886.  Idem:  Same  title,  Ibidem,  1886  (I),  49-61. 
(1888)  Idem:  Same  title,  Ibidem,  1888  (II),  142-150. 

1886.  Karl  Haushofer:  Ueber  einige  mikroskopisch-chemische  Reactionen.  Sitzb.  Akad. 
Wiss.  Miinchen,  XVI,  1886,  70-83. 

1886.  C.  Klement  et  A.  Renard:  Reactions  microchimiques  a  cristaux  el  leur  application  en 

analyse  qualitative.     Bruxelles,  1886.* 

1887.  K.  Haushofer:  Ueber  die  mikroskopischen  Formen  des  Germaniumsulfiirs  und  des 

Germaniumoxydes .    Sitzb.  Akad.  Wiss.,  Miinchen,  XVII  (1887),  133-136. 

1888.  R.  Brauns:  Miner  alien  und  Gesteine  aus  dem  hessischen  Hinterland.     Zeitschr.  d. 

deutsch.  geol.  Gesell.,  XL  (1888),  465-482,  especially  477. 

1889.  K.  Haushofer:  Ueber  eine  Methode  zum  mikroskopischen  Nachweis  von  Tantal  und 

Niob.    Sitzb.  Akad.  Wiss.,  Munchen,  XIX  (1889),  3-8. 

1889.  Erwin    Goller:  Die   Lamprophyrgdnge   des   sudlichen    Vorspessart.     Neues    Jahrb., 

B.  B.,  VI  (1889),  512  footnote. 

1890.  H.  Behrens:  Essai  d'une  methode  d' analyse  qualitative  microchimique.     Ann.  Ecole 

Polyt.  Delft.    Leiden,  VI  (1890),  82-176.* 

1890.  J.   Lemberg:  Zur   mikrochemischen    Untersuchung   einiger   Miner  ale.     Zeitschr.    d. 

deutsch.  geol.  Gesell.,  XLII  (1890),  737-752. 

1891.  K.  Zimanyi:  Ueber  Krystalle  von  Ferrisulfat.     Foldtani  Kozlony,  Budapest,  XXII 

(1891),  392.* 

1891.  H.  Behrens:  Beitrage  zur  mikrochemischen  Analyse.  Zeitschr.  f.  analyt.  Chemie, 
XXX  (1891),  125-174. 

1891.  Idem:  Reactionen  fiir  mikrochemische  Mineralanalysen.     Neues  Jahrb.,  B.  B.,  VII 

(1891),  435-470. 

1892.  K.  Haushofer:  Leitfaden  fiir  die  Mineralbestimmung,  Braunschweig,  1892.* 

1892.  B.  Frosterus:  Ueber  ein  neues  Vorkommnis  von  Kugelgranit  unfern  Wirvik  bei  Borga 
in  Finland,  nebst  Bemerkungen  uber  ahnliche  Bildungen.  T.  M.  P.  M.,  XIII 
(1892-3),  177-210,  especially  183,  footnote. 

1892.  L.  Bourgeois:  Analyse  microchimique.     Article  in  Wurtz's  Dictionnaire  de  Chimie 

Supplement  2,  Paris,  1892.  14  pp.* 

1893.  J-L.  C.  Schroeder  van  der  Kolk:  Beitrag  zur  mikrochemischen  Auffindung  von  Nickel. 

Zeitschr.  f.  wiss.  Mikrosk.,  X  (1893),  451-453. 
1893.  C.   A.   McMahon:  Notes   on  the    nicrochemical    analysis   of  rock-making  minerals. 

Mineralog.  Mag.,  X  (1893),  79-122. 
1893.  Idem:  Notes  on  the  optical  characters  of  the  globules  and  sphtrulites  of  lithium  phosphate 

and  some  other  salts.     Ibidem,  X  (1893),  229-233. 

1893.  A.  Streng:  Mikrochemische  Notizen.     Neues  Jahrb.,  1893  (I),  49-50. 

1894.  H.  Behrens:  A  manual  of  microchcmical  analysis.    London,  1894. 

1895.  Idem:  Anleitung  zur  mikrochemischen  Analyse.  Hamburg  und  Leipzig,  1895.      2nd. 

ed.,  1900.     224  pp. 

1896.  R.  Brauns:  Chemische  Mineralogie,  Leipzig,  1896.* 

1897.  R.  Brauns:  Eine  mikrochemische  Reaction  auf  Sal  peter  sdure.     Neues  Jahrb.,  1897 

(D,  73- 


ART.  511]  MICROCHEMICAL  REACTIONS  571 

1898.  W.     Florence:  Darstellung    mikroskopischer    Krystalle    in    Lothrohrperlen.     Neues 

Jahrb.,  1898  (II),  102-146. 
1900.  E.  A.  Wiilfing:  Untersuchung  des  bunten  Mergels  der  Keuperformation  auf  seine 

chemischen   und   mineralogischen    Bestandtheile.    Jahjesh.    d.    Ver.    f.  Naturk., 

Wiirttemberg,  LVI  (1900),  19-21. 

1900.  H.  Behrens:  Mikrochemische  Technik.    Hamburg  und  Leipzig,  1900,  68  pp.* 
1900.  M.   E.   Pozzi-Escot  et  H.   C.   Couquet:  Recherches  microchimiques  sur  I'yttrium, 

I 'erbium  et  le  didyme.     Comptes  Rendus,  CXXX  (1900),  1136. 

1900.  A.  C.  Huysee:  Atlas  zum  Gebrauch  bei  der  mikrochemischen  Analyse  fur  Chemiker, 

Pharmaceuten,  Berg-  und  Huttenmanner,  Labratorien  an  Universitaten  und 
technischen  Hochschuten.  Anorganischer  Teil  in  chromolithographierten  Tafeln. 
Leiden,  1900,  64  pp.,  27  pi.  (22  colored.) 

1901.  Oswald  Richter:  Ein  Beitrag  zur  Kenntnis  des  Magnesium- Ammonium-Phosphates, 

Mg  (NHJ  POt+6H20.    T.  M.  P.  M.,  XX  (1901),  89-98. 

1902.  G.  Marpmann:  Ueber  einige  neue  mikrochemische  Reaktionen.     Zeitschr.  f.  angew. 

Mikrosk.,  VIII  (1902-3),  126-130. 
1904.  Carl   Gustav  Hinrichs:  First  course  in  microchemical  analysis.     St.  Louis,    1904, 

145  PP- 

1906.  Harold  C.  Bradley:  A  delicate  color  reaction  for  copper,  and  a  microchemical  test  for 

zinc.    Amer.  Jour.  Sci.,  XXII  (1906),  326-328. 

1907.  G.    Berg:  Schneller   Nachweis   eines   Anhydritgehaltes   in  Gesteinen   und   kiinstliche 

Bildung  mikroskopischer  Anhydritkristallchen.  Centralbl.,  f.  Min.,  etc.,  1907, 
688-690. 

1908.  Fran.  Tucan:   Mikrochemische   Reaktionen   des  Gipses    und   Anhydrites.     Ibidem, 

1908,  134-136. 

1909.  Stef.  Kreutz:  Krystallisation  von  trigonalem  Silbernitrat  aus  wdsserigen  Losungen. 

T.  M.,P.  M.,  XXVIII  (1909),  488-490. 

1910.  P.   Gaubert:  Sur  la   determination  des  mineraux  par  les  reactions  colorees.     Bull. 

Soc.  Min.  France.,  XXXIII  (1910),  324-326. 

1913.  Duparc  et  Monnier:  Traite    de    technique    mineralogique   et  petrographique.     II- 1, 
Leipzig,  1913,  pp.  372. 


CHAPTER  XLI 
PREPARATION  OF  THIN  SECTIONS  OF  ROCKS 

512.  Early  History.1 — Almost  as  soon  as  the  microscope  was  known, 
attempts  were  made  to  study  the  internal  structure  of  minerals  and  rocks. 
The  first  attempts  were  made  directly  upon  the  minerals  themselves  or 
upon  chips,  as,  for  example,  when  Robert  Boyle2  in  1663,  examined  the 
inclusions  in  a  diamond  to  see  if  he  could  find  anything  peculiar  in  it.  Later 
the  rocks  were  pulverized  before  microscopical  examination,  as  by  an  un- 
known writer  in  1774, 3  and  subsequently  by  Dolomieu,4  de  Bellevue,5  and 
others.  Cordier6  improved  upon  the  method  somewhat  by  suggesting  the 
preliminary  separation  or  concentration  of  like  minerals  by  washing  or 
sliming  in  water. 

Shortly  after  the  discovery  of  polarized  light,  minerals  were  studied  by 
its  aid,  as  by  Sir  David  Brewster  in  1816  and  later,  but  while  plane-parallel 
plates  were  used  by  him  and  by  Biot,  no  systematic  attempts  were  made  to 
use  such  for  the  general  study  of  the  different  minerals.  Polarized  light 
was  made  more  available  for  the  microscope  by  Nicol's  invention,  in  1828, 
of  the  polarizing  prism  named  after  him,  and  he  prepared  thin  sections  of 
minerals.7  The  first  thin  sections  of  fossil  woods  were  thus  prepared  by  him. 
Later,  Witham,8  in  his  studies  on  fossil  woods,  made  use  of  the  same  method, 
rough  grinding  on  a  grindstone,  rough  polishing  on  a  lead  plate  with  coarse 
emery,  and  finally  on  a  copper  plate  with  fine  emery.  Sorby9  expressed 
the  opinion  that  Witham  did  not  prepare  his  own  sections  but  purchased 
them  from  Nicol,  and  also  had  some  one  else  write  his  book  for  him. 

1  See  also  F.  Zirkel:  Die  Einfuhrung  des  Mikroskops  in  das  mineralogisch-geologische 
Studium.     Decanato  Programme,  Leipzig,  1881. 

2  Robert  Boyle:  Experiments  and  considerations  upon  colours  -with  observations  on  a 
diamond  that  shines  in  the  dark.     1663.* 

3  I.  D.  in  Rozier's  Observations  sur  la  physique,  IV  (1774),  225.* 
4D.  Dolomieu:  Jour.  d.  Physique,  XLIV,  198.* 

5  Fleuriau  de  Bellevue:  Memoir e  sur  les  cristaux  microsco piques  des  laves.     Jour.  d. 
Physique,  LI  (1800),  442.* 

6  P.  Cordier:  Sur  les  substances  miner  ales  dites  en  masse  qui  entrent  dans  la  composition 
des  roches  volcaniques  de  tous  les  ages.     Ann.  Chim.  et  Phys.,  Ill  (1816),  285. 

7  H.  C.  Sorby:  Preparation  of  transparent  sections  oj  rocks  and  minerals.      Northern 
Microsc.,  II  (1882)  101-106,  133-140. 

8  Henry  Witham:  Observations  on  fossil  vegetables,  accompanied  by  representations  of 
their  internal  structure  as  seen  through  the  microscope.     Edinburgh  and  London,  1831, 48  pp.* 

Review  of  above  in  Neues  Jahrb.,  1833,  456-457. 

9  H.  C.  Sorby:  Op.  cit. 

572 


ART.  512]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  573 

This,  however,  was  the  first  published  account  of  the  process.  Slides  of 
silicified  wood  were  placed  upon  the  market  by  Andrew  Pritchard  of  London 
and  were  evidently  quite  extensively  prepared  by  him.  A  more  detailed 
account  of  the  grinding  of  sections  than  that  by  Witham  was  given  by 
Professer  linger1  of  Gratz,  who  first  sliced  his  material  on  a  stone-cutter's 
saw  and  then  ground  it  down  by  hand  with  emery  on  plates  of  bell  metal 
or  cast  iron.  His  final  polishing  was  done  by  means  of  a  circular  mo- 
tion on  damp  cloth,  tightly  stretched,  and  covered  with  tripoli  powder. 
He  said  the  thinness  of  section  necessary  for  study  must  be  such  that  fine 
print  can  be  read  through  them.  He  attached  his  chip  to  the  support  with 
a  cement  composed  of  2  parts  gum  mastic  in  grains,  4  parts  of  white  wax, 
and  i  part  of  yellow  rosin,  a  cement  which  he  claimed  to  be  better  than 
Canada  balsam,  water  glass,  shellac,  or  any  other  cementing  material, 
since  the  chip  would  not  separate  from  the  mount  except  by  heat.  The 
cover-glass  was  attached  by  means  of  Canada  balsam. 

The  first  real  use  made  of  thin  sections  was  by  Sorby 2  in  1850.  He  speaks 
of  the  preparation  of  sections  not  much  more  than  i/iooo  of  an  inch  (0.025 
mm.)  in  thickness.  If  his  sections  were  actually  of  this  thinness  they 
compared  very  favorably  with  modern  sections,  especially  since  the  rocks  he 
described  were  such,  namely  calcareous  grits,  which  ordinarily  do  not  permit 
very  thin  grinding. 

In  this,  his  first  paper  on  the  use  of  thin  sections,  Sorby  gave  no  descrip- 
tion of  the  methods  used,  and  not  much  use  was  made  of  it  by  other  investi- 
gators even  in  England,  since  two  years  later  Andrews,3  in  a  paper  before 
the  British  Association,  described  determinations  made  on  rock  splinters 
by  reflected  and  polarized  light  with  no  mention  of  any  knowledge  of  thin 
sections. 

Sorby4  continued  his  method,  however,  and  made  continual  use  of  rock 
sections  so  that,  \vith  right,  he  may  be  called  the  Father  of  Modern  Petro- 
graphic  Methods. 

In  the  meantime  the  making  of  thin  sections  had  been  taken  up  in 
Germany  by  Oschatz,5  probably  without  knowledge  of  Sorby 's  work. 
In  the  reports  of  the  meetings  of  the  German  Geological  Society  he  is  spoken 

1  Professor  Unger:    Ueber  dieUntersuchungfossilerStammeholzartigerGewdchse.     Neues 
Jahrb.,  1842,  149-171,  especially  153-159. 

2  Henry  Clifton  Sorby:    On  the  microscopical  structure  of  the  calcareous  grit  oj  the  York- 
shire coast.    Q.  J.  G.  S.,  VII  (1851),  1-6. 

See  also  Idem:  Op.  cit.,  Northern  Microsc.,  II  (1882),  101-106. 

3  T.  Andrews:    On  the  microscopic  structure  oj  certain  basaltic  and  metamorphic  rocks  and 
the  occurrence  of  metallic  iron  in  them.     Rept.  British  Asso.  Adv.  Sci.,  Belfast,  1852.     Trans- 
act, of  the  Sections,  34-35. 

4  Henry  Clifton  Sorby:    On  the  microscopical  structure  of  crystals  indicating  the  origin 
of  minerals  and  rocks.    Q.  J.  G.  S.  ,  XIV  (1858),  453-500,  especially  469. 

5  Dr.  Oschatz:    Reports  of  meetings  of  the  society.     Zeitschr.  d.  deutsch.  geol.  Gesell., 
Ill  (1851),  383;  IV  (1852),  i3;VI  (1854),  261-263;  VII  (1855),  5,  298;  VIII  (1856),  308. 


574  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  513 

of,  in  a  number  of  notices,  as  exhibiting  sections.  No  methods  of  prepara- 
tion are  given,  although  it  is  mentioned  that  most  of  the  slides  were  mounted 
in  Canada  balsam.  The  statement  is  made  that  sections  as  thin  as  i/ioo 
of  a  line  (0.0226  mm.)  were  prepared.  Collections  of  73  rock  sections  were 
offered  for  sale  at  35  Thalers  221/2  Sgr.  ($25.38)  and  separate  slides  from  one 
Thaler  (71  cents)  to  6  Sgr.  (14  cents).  Oschatz  was  the  first  man  to  at- 
tempt to  grind  minerals  soluble  in  water,  he  having  prepared  a  section  of 
carnallite  by  grinding  it  under  an  ethereal  oil.  According  to  Zirkel,1  a 
collection,  prepared  by  Oschatz,  is  in  the  University  of  Leipzig.  The  sections 
are  mounted  on  glass  slips  34X21  mm.  and  are  "extraordinarily  thin  and 
well  ground"  but  only  about  10  to  15  sq.  mm.  in  area. 

Slides  made  by  Oschatz  were  used  by  many  subsequent  investigators, 
and  more  and  more  were  thin  sections  used,  although  not  in  a  systematic  way 
until  Zirkel's2  Mikroskopischen  Gesteinsstudien  appeared  in  1863.  Zirkel  had 
himself  prepared  a  great  number  of  thin  sections  in  the  laboratory  of  the 
Geologische  Reichsanstalt  in  Vienna,  and  had  made  a  systematic  study  of  the 
material,  giving  directions  for  the  preparation  of  thin  sections  based  upon 
his  own  experience.  The  paper  was  of  great  importance,  as  was  also  a  paper 
by  him  in  i8663  in  which  he  speaks  of  his  slides  as  being  1/2  to  3/4  sq.  in. 
in  size.  Directions  for  the  preparation  of  thin  sections  were  given  by 
Vogelsang4  in  1867;  directions  which  have  been  copied,  more  or  less  word 
for  word,  by  numerous  text-books  up  to  the  present  time,  showing  how 
little  change  there  has  been  in  the  method  of  preparation. 

513.  Section-cutting  Machines. — In  preparing  thin  sections  of  rocks  the 
writer  has  found  it  advantageous  first  to  cut,  with  a  diamond  saw,  a  slice  as 
thin  as  possible  in  order  that  the  subsequent  work  of  grinding  to  the  necessary 
thinness  may  be  reduced  to  a  minimum  Certain  workers,  Forbes,5  Ady,6 
and  others,  maintain  that  it  is  more  economical  to  use  thin  chips,  doing  away 
with  all  cutting,  simply  reducing  the  chip  to  proper  thinness  by  grinding. 
Where  this  grinding  is  done  by  hand,  as  universally  seems  to  be  the  custom, 
the  value  of  the  time  lost  certainly  is  much  more  than  the  maximum  of  2/5 
of  a  cent  per  section  (two  cuts,  each  i  sq.  in.)  for  diamond  dust  used.  With 

1  F.  Zirkel:    Op.  cit.,  14,  footnote. 

2  Idem:     Mikroskopische  Gesteinsstudien.     Sitzb.  Akad.  Wiss.  Wien,  XLVII  (1863), 
226-270. 

3  Idem:    Ueber  die  mikroskopische  Zusammensetzung    und   Struktur  der   diessjahrigen 
Laven  von  Nea-Kammeni  bei  Santorin.     Neues  Jahrb.,  1866,  769-787.     With  plate  of  thin 
sections. 

4  H.  Vogelsang:    Philosophic  der  Geologic  und  mikroskopische  Gesteinsstudien.     Bonn, 
1867,  225-228. 

6  David  Forbes:  On  the  preparation  of  rock  sections  for  microscopic  examination. 
Monthly  Microsc.  Jour.,  I  (1869),  240-242. 

6  John  Ernest  Ady:  Observations  on  the  preparation  of  mineral  and  rock  sections  for  the 
microscope.  Mineralog.  Mag.,  VI  (1885),  127-133. 


ART.  513]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  575 

the  mechanical  grinder  suggested  below,  it  may  be  possible  to  equal  this  low 
cost,  but  certainly  the  time  required  to  examine  the  section  during  the 
process  would  be  worth  as  much  as  the  bort. 

There  is  not  a  great  variety  in  cutting  machines,  three  types  having  been 
made,  two  of  which  are  still  in  use.  One  of  the  earliest  instruments  was  made 
by  Rumpf.1  It  was  based  on  the  saws  used  by  stone  cutters  and  consisted 
of  a  foot-power  machine  with  a  horizontal,  hack-saw-like  arrangement  which 
was  drawn  across  the  rock.  The  blade,  however,  had  no  teeth,  was  made  of 
soft  tin  plate  stretched  taut,  and  was  fed  with  emery  and  water,  the  specimen 
being  held  against  the  blade  by  means  of  a  weight.  Slices  as  thin  as  1/2 
mm.  and  with  parallel  faces  could  be  cut  from  a  homogeneous  rock.  The 
speed  with  which  a  specimen  was  cut  depended  upon  the  kind  of  rock,  100 
sq.  cm.  of  limestone  being  sawn  in  from  three-fourths  to  three  hours,  the  same 
amount  of  granite  in  three  to  five  hours,  and  porfido  rosso  antico  in  twelve 
hours. 

Another  type  of  saw,  and  one  still  in  use  for  cutting  large  slabs,  is  also 
based  on  a  stone  cutter's  instrument.  It  consists  of  an  endless  wire  of  soft 
iron  or  brass,  0.5  to  0.7  mm.  in  diameter,  running  over  two  wheels.  The 
lower  wheel  is  attached  to  the  motive  power,  the  upper  to  a  weight  which 
serves  to  keep  the  wire  taut.  The  specimen  is  placed  on  the  sawing  table 
and  so  arranged  that  a  weight  will  draw  it  forward  mechanically  as  required. 
The  cutting  material,  emery  or  carborundum,  should  be  fed  automatically 
to  the  saw,  as  should  also  enough  water  to  keep  it  moist.  Such  a  machine 
may  be  set  working  and  left,  with  only  occasional  inspection,  until  the  cut  is 
completed.  The  speed,  in  a  granite  slab  i  in.  thick,  is  approximately 
i  1/2  in.  an  hour.  Slices  cut  with  such  a  machine  do  not  have  perfectly 
plane  faces  but  show  the  striations  made  by  the  saw  and  require  considerable 
after  grinding.  It  is  not  safe  to  attempt  slices  too  thin,  for  the  cuts  are  likely 
to  run  together.  A  preliminary  kerf  with  a  diamond  saw  is  a  great  aid  in 
starting  the  cut.  When  large  blocks  of  soft  or  medium  hard  rocks  are  to 
be  cut,  the  instrument  may  be  used  to  advantage,  but  it  is  questionable  if 
there  is  any  gain  over  the  diamond  saw  with  small  specimens  or  with  hard 
rocks,  not  only  on  account  of  its  wastefulness  of  material,  for  it  cuts  a  wide 
path  for  itself,  but  on  account  of  the  time  required. 

The  third  type  of  saw  has  a  disk-shaped  blade  revolving  On  a  spindle. 
It  may  be  either  vertical  or  horizontal  and  be  fed  with  carborundum  or  have 
its  edge  set  with  diamond  chips  or  diamond  dust. 

While  diamond  dust  was  long  used  in  the  commercial  cutting  of  certain 
stones,  its  use  in  the  laboratory  for  cutting  rock  sections  appears  to  have  been 
described  first  by  Lehmann,2  who  used  saws  made  of  tin  with  notched  edges 

1  J.  Rumpf:    Eine  Cdbinets-Steinschneide-Maschine.     T.  M.  P.  M.,  IV  (1882),  409-414. 

2  J.  Lehmann:    Einige  auf  das  Durchschneiden  von  Gesteinsstiicken  und  die  Herstellung 
von   Mineral-  und  Gesteinsdunnschlifen   beziigliche  Erfahrungen,     Verb,  naturhist.  Ver. 
preuss.  Rheinl.,  Bonn.,  XXXVII  (1880),  Sitzb.  228-231. 


576 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  513 


set  with  diamond  splinters.  About  the  same  time  Cohen  constructed, 
at  the  University  of  Strassburg,  a  rather  simple  machine  for  such  section 
cutting  but  does  not  appear  to  have  described  it.  A  modified  form  was 
made  and  described,  in  1882,  by  Steinmann,1  also  of  the  University  of 
Strassburg.  This  instrument  (Fig.  746)  is  very  similar  to  modern  instru- 
ments and,  with  the  exception  of  more  delicate  adjustments  on  some  of  them 
for  accurate  orientation  of  crystals,  serves  as  a  model  for  modern  makers. 
The  instrument  is  worked  by  foot  power  and  is  about  the  size  of  a  large 
sewing  machine.  The  top  (M)  is  made  of  wood,  covered  by  a  sheet  of  zinc 
(T)  which  slopes  to  the  left-hand  rear  corner  so  that  all  water  spilled  upon  it 
will  drain,  through  the  tub.e  a,  to  a*  pan  set  on  the  floor.  P  is  a  cast-iron  plate 

to  which  are  attached  the  tracks  5*5 
and  Si,  carrying  the  guiding  apparatus 
parallel  to  the  cutting  saws  s  and  SL  Si 
may  be  displaced  or  removed  by  the 
screws  JJL  and  //i.  The  carrier  at  the  left 
may  be  shoved  forward  by  the  plate  B, 
and  the  specimen  inclined  at  any  angle 
by  means  of  the  plates  C  and  D.  D 
has,  at  the  back,  a  slit  for  the  en- 
trance of  the  saw.  Attached  to  F  is  a 
horizontal  plate  with  a  F-shaped  cut- 
out, likewise  for  the  entrance  of  the 
saw  when  the  plate  FF  is  inclined  in 
azimuth  by  means  of  the  screws  d  and  e. 
At  the  right,  the  sledge  BI  carries  the 
bar  Ji  to  which  the  cylinder  NI  is  at- 
tached. In  the  latter  is  the  T-shaped 
bar  CiDi.  The  bar  D\  is  hollow  and 
carries  the  plate  FI,  which  may  be 
shoved  forward  by  means  of  the  screw 

7  and  clamped  by  f .  To  the  axle  A  are  attached  the  two  saws  s  and  s\ 
which  are  turned  by  means  of  the  pulley  R.  The  bottle  G  contains  the  petro- 
leum which  is  used  as  a  lubricant  and  is  conducted  to  the  saws  by  means  of 
the  lead  tubes  r  and  ri,  which  may  readily  be  bent  to  proper  positions.  The 
tin  shields  H-H,  whose  front  halves  Hi-Hi  may  be  thrown  back,  serve  as  mud 
guards,  and  are  provided  with  windows/  and/i  through  which  the  process  of 
cutting  may  be  observed. 

The  saws  of  tin  plate  (tinned  iron)  used  by  Steinmann  were  from  10  to  22 
cm.  in  diameter.  The  central  holes  were  made  a  trifle  smaller  than  the  counter 
shaft  upon  which  they  were  to  be  placed,  and  were  enlarged  by  means  of  a 

1  Gustav  Steinmann:  Eine  verbesserte  Steinschneidemaschine.  Neues  Jahrb.,  1882 
(II),  46-54- 


FIG.  746. — Steinmann's  section  cutting  machine. 


ART.  513] 


PREPARATION  OF  THIN  SECTIONS  OF  ROCKS 


577 


rat-tail  file  until  they  fitted  exactly.  After  being  clamped  to  the  shaft,  they 
were  rotated  against  a  hard  substance,  such  as  the  flat  edge  of  a  smoothly 
ground  triangular  file,  until  no  more  shavings  were  removed,  and  they  were, 
consequently,  perfectly  true  and  had  edges  at  right  angles  to  the  sides. 
The  edge  was  now  nicked  and  charged  with  diamond  dust.1 


FIG.  747. — Section  cutting   and   grinding   apparatus.     (Dr.  Steeg  and  Reuter.) 

The  rock  to  be  cut  is  attached  to  the  plate  by  a  prepared  cement  made  of 
i  part  beeswax  and  i  part  yellow  rosin.  For  small  sections  the  Si  saw  is 
used,  the  specimen  being  fastened  to  F\.  If  the  chip  has  no  flat  surface,  it 
is  set  in  wax  and  further  supported  by  burnt  matches  stuck  into  wax-filled 
holes  in  the  plate  F\.  These  small  wooden  supports  are  cut  by  the  saw  as  it 
passes  through  the  rock.  Should  the  plate  Si  be  not  quite  vertical,  it  may  be 
adjusted  by  placing  thin  sheets  of  paper  or  card-board  under  one  side  or 
the  other  of  its  base.  To  test  whether  the  saw  is  perfectly  vertical,  saw  a 
thick  piece  of  a  soft  rock,  such  as  a  homogeneous  limestone,  fasten  the  newly 
sawed  face  of  the  piece  removed  to  the  plate,  and  make  a  new  cut.  The 

1  See  Art.  514,  infra. 
37 


578 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  513 


difference  in  the  thickness  of  top  and  bottom  will  represent  the  departure 
from  parallelism  between  saw  and  holder  plate.  To  test  whether  the  plate 
FI  is  parallel  to  the  saw  in  azimuth,  measure  the  front  and  back  of  the  soft 
rock  just  cut.  The  thickness  should  be  the  same.  If  not,  correct  it,  and 
then  cut,  with  a  file,  a  scratch  across  Ci  and  NI  to  mark  this  position.  For 
mineralogical  work  it  is  advantageous  if  the  top  of  the  cylinder  NI  is  marked 
in  2°  divisions.  The  instrument  was  made  complete,  by  Carl  Benz  in  Mann- 
heim, for  275  Marks. 

Made  somewhat  on  the  same  principle,  although  with  but  a  single  saw, 
are  the  cutting  machine  first  described  by  Groth1    in  1885,    and  the  one 


FIG.  748. — Hand  section  cutting  machine.     (Dr.  Steeg  and  Reuter.) 


shown  in  Fig.  747.  Both  machines  are  made  for  combined  cutting  and 
grinding.  In  the  one  shown  in  the  illustration,  the  specimen  to  be  cut  is 
cemented  to  the  plate  a,  which  may  be  moved  along  the  rod  b  for  a  con- 
siderable distance.  At  the  end  of  this  rod  are  two  screws  (0)  for  fine  adjust- 
ment, permitting  the  cutting  of  a  section  of  any  thickness.  The  weight  c 
may  be  removed  up  or  down  the  rod  to  which  it  is  attached,  thus  changing  the 
pressure  against  the  saw  d,  which  rotates  from  the  top  downward.  For 
large  specimens  the  clamp  n  is  used  instead  of  the  disks  a  and  d.  Should 
the  saws  become  eccentric  they  may  be  trued  up  with  a  turning  chisel, 
using  the  object  carrier  as  a  tool  rest. 

1  P.  Groth:  Physikalische  Krystallographie.    Leipzig,  2  Aufl.,  1885,  670. 


ART.  513] 


PREPARATION  OF  THIN  SECTIONS  OF  ROCKS 


579 


Another  instrument  described  by  Groth1  was  a  small  hand  apparatus, 
similar  to  the  one  shown  in  Fig.  748  except  that  the  power  was  transmitted 
by  geared  wheels  instead  of  a  belt.  The  orienting  device  shown  at  the  side 
is  extremely  useful  in  cutting  sections  along  certain  definite  directions.  A 
similar  device  is  described  by  Fuess.2 

The  cutting  machine  shown  in  Fig.  749  has  one  decided  advantage  over 
those  previously  described  in  its  extremely  rigid  holder  (K-S)  for  the  speci- 
men. This  carrier  is  mounted  on  a  rod  and  may  be  moved  very  accurately 
to  any  required  distance,  by  means 
of  the  screw-head  s,  in  a  direction 
at  right  angles  to  the  plane  of  the 
saw.  The  latter  has  a  diameter  of 
6  1/2  to  7  in.  (16-18  cm.)  and 
should  be  rotated  at  a  speed  of 
about  400  revolutions  per  minute, 
the  power  being  applied  either  by 
foot  or  motor.  The  special  recom- 
mendation for  this  instrument  is  its 
compactness,  the  freedom  of  the 
saw  from  vibration,  and  the  rigidity 
of  the  specimen  carrier.  If  the  saw 
becomes  eccentric,  as  it  is  likely  to 
do  when  carborundum  and  not  dia- 
mond dust  is  used  with  it,  it  may 
be  trued  up  by  swinging  out  the 
carrier  K  and  using  it  as  a  rest 
for  a  turning  tool.  In  a  sawing 
machine  described  by  Rauff 3  a  turn- 
ing tool  could  be  clamped  in  the 
specimen  carrier  and  advanced  by 
means  of  a  cranked  screw  similar  to 
that  in  a  machine  lathe. 

The  cutting  apparatus  described  by  Grayson4  differs  from  those  pre- 
viously described  in  a  number  of  particulars.  The  ordinary  machines  run 

1  P.  Groth:  Op.  cit.,  668. 

2  R.  Fuess:    Ueber  eine  Orientirungs-oorrichtung  zum  Schneiden  und  Schleifen  von  Miner- 
alien  nach  bestimten  Richtungen.     Zeitschr.  f.  Instrum.,  Oct.  4,  1899,  4  pp.  separate. 

3  H.  Rauff:    Ueber  eine  verbesserte  Steinschneidemaschine,  sowie  ilber  einen  von  M.  Wolz 
in  Bonn  construirten  damit  verbundenen  Schleif-Apparat  zur  Herstellung  genau  orientirter 
Krystalplatten.     Neues  Jahrb.,  1888  (II),  230-246. 

Idem:  Same  title  in  Verhandl.  d.  naturhist.  Vereins.  d.  preuss.  Rheinlande  u.  West- 
falens.  1886,  130-139.* 

4  H.  J.  Grayson:    Modern  improvements  in  rock-section  cutting  apparatus.     Proc.  Roy. 
Soc.  Victoria,  XXIII,  Pt.  I  (1910),  65-81. 


FIG.  749. — Section  cutting   machine.     (Fuess.) 


580  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  514 

best  at  about  500  revolution  per  minute  with  disks  8  in.  or  less  in  diameter, 
while  Grayson's,  with  a  disk  10  in.  in  diameter,  is  speeded  to  1000  revolutions. 
Instead  of  being  vertical,  the  cutting  disk  is  horizontal,  and  is  clamped 
between  two  collars.  With  his  instrument  he  claims  to  be  able  to  slice, 
grind,  and  mount  a  granite  section  an  inch  in  diameter,  in  not  more  than 
ten  minutes.  The  cost  of  charging  a  lo-in.  disk  with  diamond  dust  is  one 
shilling,  and  with  it  he  is  able  to  cut,  without  recharging,  about  95  sq.  in. 
of  average  rock,  making  the  cost  about  1/4  cent  per  square  inch. 

514.  Diamond  Saws. — Diamond  saws  are  of  three  kinds,  those  set  with 
diamond  chips,  those  charged  with  diamond  dust  directly  upon  the  smooth 
edge  of  a  disk,  and  those  charged  in  notches.  The  first  kind  is  made  by 
setting  the  chips  in  the  edge  of  a  tin  disk  much  in  the  manner  of  the  setting 
of  writing  diamonds,  namely  by  gouging  out  a  hole,  inserting  the  chip,  and 
burnishing  down  the  burr.  The  saws  are  expensive,  costing,  at  the  present 
price  of  bort,  about  $9.50  for  a  y-in.  saw,  and  $13.50  for  one  10  in.  in  di- 
ameter. For  the  purpose  of  cutting  rock  slices  they  are  not  nearly  so  satis- 
factory as  disks  charged  with  diamond  dust.  The  makers  claim  that  with 
proper  usage  a  y-in.  saw  will  cut  about  470  sq.  in.  of  rock  of  average  hardness, 
making  the  cost  about  2  cents  per  square  inch.  As  ordinarily  used  they  cut 
decidedly  less.  Aside  from  the  expense,  the  fact  that  if  slightly  eccentric 
they  cannot  be  turned  true,  is  a  serious  objection  to  them. 

The  second  class  of  saws  are  those  charged  directly  upon  the  edge  of  a 
smooth  disk  of  ordinary  tin  plate,  about  0.50  to  0.75  mm.  in  thickness.  The 
disk  should  be  from  6  to  10  in.  in  diameter;  a  7-in.  saw  being  quite  satis- 
factory for  ordinary  work.  The  smaller  the  saw,  the  more  rigid  it  is, 
consequently  the  truer  it  will  run.  The  large  saws  are  chiefly  of  use  in  cut- 
ting slabs  from  large  blocks.  The  depth  of  cut  which  can  be  made  will 
depend  upon  the  size  of  the  counter-shaft  and  the  nut  and  bearing  plates 
holding  the  saw.  Usually  the  cut  will  be  about  one-third  the  diameter  of 
the  disk.  By  reversing  the  specimen,  a  piece  somewhat  less  than  two- 
thirds  the  diameter  of  the  saw  may  be  cut. 

A  saw  should  run  perfectly  true  and  with  no  eccentricity.  The  central 
hole  should  be  a  trifle  smaller  than  the  counter-shaft,  and  the  disk  should  be 
fitted  as  described  by  Steinmann.1  If  there  is  any  eccentricity,  it  should 
be  removed  by  turning  oil  the  edge  by  an  ordinary  metal- turning  chisel, 
using  the  rock  holder  as  a  tool  rest,  or  even  clamping  the  chisel  in,  if  the 
holder  is  suitable  for  so  doing.  The  edge  of  the  saw  should  be  exactly  at 
right  angles  to  the  sides,  otherwise  it  will  cut  in  at  an  angle  and  soon  bind. 

To  charge  the  disk  the  following  method  is  recommended  by  Leiss.2  In 
a  fragment  of  quartz  or  flint,  a  cut,  5  to  10  mm.  in  depth,  is  made,  and  in  it  is 
placed  a  very  small  quantity  of  diamond  dust  moistened  with  petroleum. 

'Art.  513. 

2  C.  Leiss:  Die  optischen  Inslrumente,  etc.      Leipzig,  1899,  274. 


ART.  514]  PREPARATION  OF  THIX  SECTIONS  OF  ROCKS  581 

The  saw  kerf  is  slipped  over  the  edge  of  the  tin  disk  and  is  pressed  hard 
against  it.  The  belt  or  pulley  which  turns  the  spindle  is  now  rotated  by 
short  forward  and  backward  motions  until  the  entire  disk  has  passed  through 
the  quartz.  By  this  means  the  diamond  dust  is  forced  into  the  rim  and 
the  adjacent  sides  of  the  tin. 

Another  method  of  charging  the  saw  is  to  apply  a  thin  paste  of  diamond 
dust  and  olive  oil  to  the  edge  of  the  rapidly  rotating  disk  by  means  of  a  very 
small,  spatula-shaped  piece  of  wood,  and  at  the  same  time  hold,  with  the 
other  hand,  a  smooth  piece  of  quartz  or  agate,  moistened  with  petroleum, 
against  the  running  edge.  The  whole  process  of  charging  requires  but  a  few 
moments  and  but  a  very  small  quantity  of  diamond  dust  is  used. 

The  objection  made  to  smooth-edge  saws  has  usually  been  that  when  rocks 
which  are  composed  of  minerals  of  different  hardness  are  cut,  the  diamond 
cutting-edge  is  quickly  lost.  Grayson1  seems  to  have  found  them  satis- 
factory for  general  use.  For  perfectly  homogeneous  material,  such  as  agate, 
the  saw  is  excellent  and  will  cut  from  25  to  35  sq.  in.  at  a  cost  of  about  one- 
fourth  of  a  cent  per  square  inch.  When  the  saw  becomes  dull  it  may  be 
recharged  in  the  same  manner  as  before.  If  it  becomes  eccentric  it  is  simply 
necessary  to  turn  it  true,  and  recharge. 

The  third  type  of  saw  is  the  tin  disk  with  nicked  edge.  Such  a  saw  was 
first  used  for  petrographical  sections  by  Steinmann2  in  1882.  He  used  disks 
of  from  10  to  22  cm.  in  diameter,  and  with  a  knife  hacked  the  edges  in  a 
tangential  rather  than  a  radial  direction,  thus  making  notches,  rip-saw-like, 
close  together  all  around  the  rim.  The  saw  is  so  placed  on  the  counter- shaft 
that  the  notches  point  downward  in  front,  thus  forcing  the  diamond  dust 
deeper  as  the  saw  rotates  from  the  top  forward.  The  saw  is  charged  by 
mixing  diamond  dust  with  kerosene  or  other  oil  to  form  a  thick  paste,  and 
placing  it  on  the  edge  of  the  saw  at  intervals  of  one-sixth  to  one-eighth  the 
circumference.  As  little  as  possible  should  be  placed  on  the  sides  for  there 
the  material  is  practically  wasted.  The  pulley  should  now  be  rotated  by 
hand  while  a  smooth  piece  of  quartz  is  pressed  against  the  edge  of  the  disk 
to  force  the  diamond  dust  into  the  soft  iron.  More  dust  is  now  placed  at 
intervals  around  the  edge  and  the  rotation  repeated,  and  thus  the  process 
is  continued  until  the  entire  disk  has  been  charged.  If  any  of  the  paste 
adheres  to  the  side  of  the  saw  it  may  be  recovered  by  washing  it  into  a  beaker 
by  means  of  oil  from  a  small  wash  bottle;  the  diamond  dust  soon  settling  to 
the  bottom  of  the  beaker.  When  a  saw  is  dull  it  may  be  recharged  as  before. 
If  eccentric  it  may  be  turned  circular,  rehacked,  and  recharged. 

Recent  use  seems  to  indicate  that  radial  nicks  are  more  efficient  than 
tangential.  After  truing  up  a  saw,  hack  its  edge,  by  means  of  an  old,  rather 
heavy,  thin-bladed  case  knife,  with  a  force  sufficient  to  make  incisions  about 

1  H.  J.  Grayson:  Op.  cit.,  p.  76. 

2  Gustav  Steinmann:  Op.  cit. 


582  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  514 

a  millimeter  in  depth.  Rotate  the  disk  by  hand  until  the  entire  edge  is 
covered  with  nicks  about  a  millimeter  apart.  Even  distribution  in  the 
incisions  is  not  at  all  essential  although  it  makes  a  better  looking  saw.  The 
edge  may  be  charged  with  diamond  dust  in  the  manner  described  above,  or 
it  may  be  rubbed  into  the  edge  with  the  finger,  or  the  disk  may  be  removed 
from  the  counter-shaft  and  the  edge  placed  in  a  little  heap  of  the  oil  and 
diamond  dust  placed  on  a  flat  iron  plate  and  the  upper  edge  gently  hammered 
while  rotating  the  disk.  To  be  certain  that  it  is  completely  charged,  it 
should  be  thus  rotated  two  or  three  times  through  the  powder.  After  charg- 
ing the  edge,  a  smooth  piece  of  quartz  or  a  chilled  steel  roller  is  held  against 
the  edge,  very  gently  at  first,  to  force  the  chips  deeper  into  the  iron. 

The  diamond  used,  if  possible,  should  be  purchased  as  coarse  bort  and 
not  in  powdered  form.  It  should  be  crushed  in  a  steel  diamond  mortar  and 
sifted  during  the  process  so  that  the  material  obtained  is  of  quite  uniform 
size.  Grayson1  made  sieves  out  of  inch-long  sections  of  3/4-in.  glass  tubing, 
to  one  end  of  which,  after  grinding  level,  a  piece  of  very  fine  bolting  silk  was 
cemented.  One-forth  of  a  karat  (1/20  grm.)  of  dust  is  sufficient  to  charge  a 
half  dozen  6-in.  saws,  and  costs  about  35  to  40  cents,  making  the  cost  per 
saw,  including  tin,  less  than  ten  cents. 

Saws  should  run  in  perfectly  true  planes.  They  may  be  tested  for  ad- 
justment as  described  in  Art.  513.  Upon  this,  and  upon  lack  of  eccentricity, 
much  of  the  efficiency  depends.  They  should  not  be  used  as  long  as  they  will 
cut,  but  should  be  recharged  whenever  they  begin  to  be  dull.  One  can  make 
his  own  disks  by  cutting,  with  a  tin  shears,  an  approximately  circular  disk 
from  a  piece  of  perfectly  flat  tin,  and  then,  placing  it  on  the  spindle, 
truing  it  up  with  a  turning  tool.  It  is  well  to  cut  and  charge  a  number  of 
saws  at  the  same  time,  so  that  they  may  be  on  hand  when  wanted. 

A  saw  should  run  from  above  downward,  and  during  the  process  of  cutting 
it  should  be  kept  constantly  wet  with  a  lubricant  of  some  kind.  Kerosene, 
sweet  oil,  water,  sodium  carbonate  and  water,  and  soap  emulsion  have  all  been 
recommended.  The  first  is  probably  best  for  compact  rocks  and  the  last 
for  those  which  are  soft  or  porous.  Sweet  oil  is  likely  to  become  sticky  and 
is  hard  to  remove  from  the  specimen.  Some  sort  of  reservoir  placed  above 
the  cutting  bench,  and  with  a  dropping  tube  conducting  to  the  upper  edge 
of  the  saw,  is  most  convenient.  It  should  be  so  adjusted  that  two  or  three 
drops  fall  per  second,  the  right  amount  depending  somewhat  upon  the 
character  of  the  rock,  but  is  easy  to  determine  since  it  must  be  great  enough 
to  keep  the  saw  wet  and  not  enough  to  spatter  a  great  deal.  A  good  sugges- 
tion, made  by  Grayson,  is  to  attach  two  bits  of  sponge  beneath  the  drip  and 
so  arranged  that  one  piece  touches  either  side. 

The  size  of  the  saw  to  be  used  depends  upon  the  nature  of  the  work  and 
the  size  of  the  specimen.  The  smaller  the  saw,  the  truer  will  it  run.  If  a 

1  H.  J.  Grayson:  Op.  cit.,  p.  75. 


ART.  515]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  583 

large  saw  is  used,  it  is  well  to  make  a  preliminary  cut,  5  to  10  mm.  deep, 
with  a  small  one,  and  start  the  larger  saw  in  the  kerf.  All  through  the  opera- 
tion, but  especially  at  the  beginning,  the  specimen  should  be  but  lightly 
pressed  against  the  saw.  Only  a  certain  amount  of  material  can  be  removed 
by  each  diamond  chip,  and  pressure  does  not  make  this  any  greater.  The 
number  of  rotations  depends  upon  the  size  of  the  saw.  Usually  400  to  600 
revolutions  are  considered  sufficient  for  a  y-in.  saw,  and  less  for  those 
that  are  larger,  although  Gray  son  recommends  a  speed  of  1000  revolutions 
for  a  lo-in.  saw.  The  higher  the  speed,  the  more  true  must  the  saw  run. 

The  rapidity  with  which  a  saw  will  cut  a  section  depends  entirely 
upon  the  rock.  With  a  saw  in  fair  condition,  a  square  inch  of  chalcedony 
should  be  cut  in  about  three  minutes ,  granite  in  two,  and  syenite  in  one. 

The  life  of  a  y-in.  notched  saw  is  from  50  to  60  sq.  in.  of  average  rock. 
Very  soft  rocks  should  not  be  cut  with  the  diamond  saw,  but  with  a  circu- 
lar disk  fed  with  carborundum.  Rocks  which  consist  of  minerals  of  very 
different  degrees  of  hardness  should  be  moved  forward  very  slowly. 

515.  Sawing  a  Rock  Slice. — The  method  of  sawing  a  rock  slice  has  not 
altered  very  much  since  the  first  one  was  cut  by  Trautz,  as  described  by 
Steinmann.1  If  one  has  brought  chips  as  well  as  hand  specimens  from  the 
field,  the  former  may  be  taken,  ground  to  a  plane  surface  on  one  side  with 
carborundum  upon  a  lap,  and  cemented  to  the  receiving  plate  (a,  Fig.  747, 
K,  Fig.  749).  If  a  number  of  sections  are  to  be  made,  three  or  four  chips 
from  rocks  of  approximately  the  same  degree  of  hardness  may  be  cemented 
to  the  plate  at  the  same  time.  If  one  grinds  his  own  sections,  he  will  soon 
learn  to  bring  from  the  field  chips  which  are  thin,  nearly  plane  on  one  side, 
free  from  cracks,  and  about  i  1/2  in.  in  diameter. 

Different  workers  prefer  different  cements.  One  of  the  most  common  is 
Canada  balsam,  which,  however,  requires  previous  boiling.2  Steinmann's 
cement  of  wax  and  rosin  is  very  good  when  properly  prepared.  If  too  soft 
the  section  will  glide  in  it,  if  too  hard  the  cement  will  fracture.  An  ex- 
cellent cement  is  common  shellac,  or  half  shellac  and  half  Canada  balsam. 
This  should  be  used  with  a  heated  plate  and  a  heated  specimen.  Like 
chips  mounted  in  balsam  alone,  they  may  be  removed  by  heat  or  by 
placing  them  for  several  hours  in  alcohol.  If  no  chip  was  brought  from  the 
field  or  if  they  are  too  large  or  irregular  to  be  readily  ground  to  a  flat 
surface  on  one  side,  or  if  it  is  desired  to  make  a  section  transversely  across 
a  schistose  rock,  it  will  be  necessary  to  make  a  preliminary  saw  cut  by 
clamping  the  specimen  in  the  holder  provided  and  sawing  off  the  desired 
amount.  Usually  such  pieces  will  be  found  to  have  surfaces  flat  enough  to 
be  cemented  directly  to  the  holder  plate,  although  it  may  be  a  wise  precaution 
to  give  them  a  few  rubs  on  the  lap  to  assure  a  flat  surface.  If  this  face  is 

1  Gustav  Steinmann:  Op.  cit. 

2  See  Art.  519. 


584  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  515 

to  be  the  side  attached  to  the  permanent  mount,  it  must,  of  course,  be  care- 
fully ground. 

After  cementing  the  chips  to  the  disk,  the  holder  is  screwed  up  until  the 
saw  is  at  the  desired  distance  from  the  plate.  The  thinness  to  which  a 
rock  may  safely  be  sawed  depends  upon  its  character.  Compact  homo- 
geneous rocks  may  be  made  much  thinner  than  those  which  are  porous, 
0.5  to  i.o  mm.  being  an  average  cut.  If  a  piece  of  tin  of  the  proper  thickness 
is  laid  against  the  face  of  the  holder  plate  adjacent  to  the  mineral  chips,  the 
holder  may  be  pushed  forward  until  the  two  rest  against  the  saw.  Leaving 
the  tin  in  position  when  the  saw  is  started,  the  slide  will  be  of  the  proper 
thickness.  Instead  of  a  tin  strip  the  writer  recommends  a  screw  with  a 
flat  end  and  extending  through  the  holder  plate  near  the  rear  edge.  This 
may  be  made  to  project  the  proper  distance  and  serve  as  a  guide.  Care  must 
be  used  to  let  the  screw  touch  the  side  of  the  saw  and  not  let  the  diamond- 
charged  edge  slice  off  the  end. 

Toward  the  end  of  a  cut,  the  pressure  against  the  saw  should  be  decreased, 
although  the  speed  of  rotation  should  remain  the  same.  The  hand  should  be 
held  against  the  fragment  being  cut  off,  otherwise  its  weight  may  tear  away 
a  piece  of  the  specimen. 

If,  at  any  time  during  the  cutting,  the  rock  tears  loose  from  the  mount, 
refasten  it  immediately  by  heating.  Upon  starting  up  the  saw,  begin  anew 
from  the  opposite  side  and  meet  the  former  cut,  but  do  not  proceed  in  the 
old  kerf. 

Sometimes  a  saw  will  bind.  This  is  due  to  a  deflection  from  the  true 
plane  of  the  saw,  and  may  be  caused  by  having  pressed  forward  too  rapidly 
during  the  process,  by  a  lack  of  lubricant,  or  by  a  false  start.  Remove  the 
saw  and  make  a  fresh  cut  to  meet  the  old.  If  there  are  a  few  very  hard 
minerals,  such  as  garnets,  scattered  through  the  rock,  the  pressure  which  forces 
the  saw  forward  should  be  decreased,  otherwise  upon  meeting  a  face  of  the 
hard  mineral  at  a  very  small  angle,  the  saw  will  be  forced  aside  and  bind. 

Having  cut  the  rock,  the  slice  may  be  removed  from  the  iron  supporting 
plate  by  heating.  Instead  of  mounting  the  chips  directly  on  the  iron  sup- 
porting plate,  it  is  sometimes  desirable  to  mount  them  upon  a  glass  support 
which  later  serves  as  a  holder  for  grinding.  This  is  usually  desirable  when 
the  rocks  are  too  brittle  to  support  themselves  in  thin  slices.  In  such  cases, 
after  the  preliminary  cut,  the  flat  side  is  ground  with  particular  care  to  a  flat 
surface  as  described  below,  and  mounted  on  i  i/4-in.  squares  of  plate  glass,  or 
even  on  ordinary  object-glasses.  The  lower  sides  of  these  glass  slips  are  now 
fastened  to  the  iron  supporting  plate  with  beeswax,  or  a  beeswax  and  rosin 
cement,  the  former  being  the  easier  to  remove,  and  the  cutting  is  done  as 
before.  With  whatever  cement  the  glass  slip  is  attached  to.  the  plate,  it 
should  have  a  lower  melting-point  than  the  Canada  balsam  with  which  the  slice 
is  mounted  to  the  object-glass,  so  that  it  will  loosen  first. 


ART.  516]  PREPARATION  OF  THIX  SECTIONS  OF  ROCKS  585 

516.  Grinding  a  Section. — Before  making  a  saw  cut  it  is  sometimes  neces- 
sary, as  was  mentioned  above,  to  grind  upon  the  chip  one  perfectly  plane  sur- 
face. This  may  be  done  by  hand  on  a  metal  or  glass  plate  covered  with  wet 
emery  or  carborundum,  or  it  may  be  done  upon  a  rotating  lap.  The  old 
method  of  grinding  by  hand  is  still  employed  by  some  men  in  preference  to 
using  a  machine,  but  it  requires  more  time  and  certainly  need  not  be  em- 
ployed until  the  final  stage  of  grinding  the  second  face  is  reached. 

For  grinding  it  is  necessary  to  have  several  grades  of  emery  or  carborun- 
dum. The  writer  uses  No.  120  carborundum  for  coarse  grinding,  follows 
with  No.  1 80,  and  concludes  with  FFFF  emery.  For  the  very  last  grinding 
it  is  better  to  use  old  emery  rather  than  that  which  is  fresh  and  unused. 
Extreme  care  must  be  taken  not  to  mix  the  coarse  with  the  fine,  a  single 
grain  of  coarse  carborundum  on  the  final  plate  may  cut  a  slide  to  pieces  in 
an  instant. 

For  hand  grinding,  three  plates  of  metal  or  glass  should  be  used.  Ady1 
recommends  plates  of  soft  metal  such  as  pewter,  zinc,  copper,  or  lap-metal, 
each  about  12  in.  square  and  1/4  in.  thick,  and  says  that  glass  plates  are 
"not  to  be  tolerated"  since  they  rapidly  wear  down  irregularly.  He  says 
further  that  the  plates  should  be  raised  slightly  in  the  center  so  that  the 
slide,  from  the  natural  greater  pressure  on  the  edges,  will  not  be  thicker  in 
the  center.  By  heating  a  metal  plate  in  the  center,  by  means  of  the  Bunsen 
burner,  it  will  buckle  by  expansion  and  give  enough  curvature.  It 
should  then  be  set  in  a  wooden  frame  on  a  base  of  plaster  of  Paris.  Zirkel2 
and  Rosenbusch3  recommend  a  plate  of  cast  iron  for  the  rough  grinding,  and 
one  of  ground  glass  for  the  fine.  The  writer  has  used  pieces  of  plate  glass 
for  coarse  and  fine,  and  has  had  no  difficulty  with  their  grinding  away  un- 
evenly, but  when  used  by  students  they  do  so  rapidly.  A  little  care  is 
necessary  and  all  parts  of  the  plate  should  be  used.  Nor  is  there  any  need 
of  a  plate  with  a  raised  center  when  one  has  had  a  little  experience. 

Very  little  grinding  is  necessary  before  moun  ing  if  the  chip  has  been 
sawed  and  is  to  be  mounted  on  the  iron  support  of  the  cutting  machine,  or 
if  it  is  to  be  mounted  on  a  piece  of  thick  glass  for  the  first  grinding  and 
is  to  be  attached  to  the  final  object  slip  by  the  second  sawed  face.  If 
the  final  grinding  is  done  on  the  heavy  glass  and  the  slice  is  transferred  by 
floating  to  the  new  glass,  the  first  face,  of  course,  must  be  ground  with  fine 
emery. 

Upon  one  plate  is  placed  a  pinch  of  No.  120  carborundum  which  is  w.et, 
and  kept  wet,  with  considerable  water;  more  water  or  more  carborundum 
being  added  as  either  seems  to  be  needed.  Now,  with  a  circular  motion, 

1  John  Ernest  Ady:    Observations  on  the  preparation  of  mineral  and  rock  sections  for  the 
microscope.     Mineralog.  Mag.,  VI  (1885),  125-133. 

2  F.  Zirkel:    Lehrbnch  der  Petrographie.  I,  Leipzig,  1893,  21. 
3Rosenbusch-Wiilnng:    Mikroskopische  Physiographie,  I-i,  Stuttgart,  1904,  108. 


586  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  516 

first  in  one  quarter  of  the  plate  and  then  in  another,  and  with  occasional 
sweeps  over  its  entire  surface,  the  slide  is  rubbed  until  it  has  become  perfectly 
flat.  If  a  chip  and  not  a  sawed  face  was  used  to  begin  with,  grind  until 
there  appears  a  flat  surface  an  inch  in  diameter  without  depressions.  Wash 
well  in  water,  using  an  old  tooth  brush  to  remove  all  grains  of  the 
coarse  emery,  and  begin  grinding  in  the  same  way  on  the  second  plate  upon 
which  is  placed  water  and  a  pinch  of  No.  180  carborundum.  Grind  until  no 
scratches  remain,  and  be  sure,  by  applying  equal  pressure  over  the  entire 
surface  while  grinding,  that  the  face  is  perfectly  flat. 

Again  wash  the  slice  carefully  with  water,  and  transfer  it  to  the  third 
plate  upon  which  is  placed  some  water  and  a  little  FFFF  emery.  Here 
grind  till  no  traces  of  marks  from  the  second  emery  remain.  It  is  not 
necessary  to  polish  the  surface,  the  process  being  more  than  likely  to  spoil 
the  flatness  of  the  face. 

Wash  again,  and  mount  the  face  just  prepared  in  Canada  balsam  upon 
the  final  object-slip,  and  at  the  same  time,  while  still  warm,  slip  off  the  heavy 
glass  base,  if  such  was  used.  Press  down  firmly  until  the  balsam  is  cool, 
and  then  repeat  the  process  of  grinding  upon  the  other  face.  The  most 
delicate  part  of  the  operation  is  the  final  grinding  after  the  slide  has  reached 
almost  its  proper  thinness.  No  amount  of  written  instructions  can  teach  the 
proper  time  to  change  from  the  second  grade  of  carborundum  to  the  emery, 
nor  when  new  emery  may  no  longer  be  safely  added.  This  must  be  learned 
by  experience.  Do  not  expect  to  make  a  section  at  the  first,  or  second, 
or  third,  trial.  After  five  or  ten  you  may  succeed  and,  thereafter,  the  knack 
of  keeping  the  slide  absolutely  level  on  the  grinding  plate  having  been 
obtained,  the  work  is  easy. 

When  a  slice  of  a  dark  rock  is  thin  enough  to  permit  fine  print  to  be  read 
through  it,  it  should  be  covered  with  a  drop  of  water  and  the  interference 
colors  examined  under  the  microscope  to  determine  its  thickness.  Any  old 
microscope,  abandoned  in  the  laboratory  as  out  of  date,  or  a  preparation 
microscope  (Figs.  723  and  724)  may  be  used.  Rocks  which  contain  consider- 
able quartz,  such  as  granite,  will  be  transparent  long  before  they  are  of  suffi- 
cient thinness.  Print  may  first  be  read  through  the  quartz,  then  through  the 
orthoclase,  then  through  the  albite,  and  finally,  perhaps,  through  the  ferro- 
magnesian  minerals.  After  grinding  a  number  of  sections  the  proper  degree 
of  thinness  will  be  learned  by  noting  how  the  slide  is  acting  in  regard  to 
breaking  away  at  the  edges.  A  good  section  should  be  between  0.025  and 
0.040  mm.  in  thickness  so  that  quartz  will  show  an  interference  color  no  higher 
than  yellow.  Be  very  careful  not  to  grind  too  long  on  the  No.  180  carborun- 
dum before  transferring  to  the  emery,  for  you  may  have  a  square  inch  of 
rock  one  moment  and  the  next,  with  but  a  single  sweep  across  the  plate,  it 
has  disappeared.  It  is  better,  toward  the  end,  to  place  the  slide  often  under 
the  microscope  and  examine  the  interference  colors. 


ART.  516]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  587 

If  the  microscopic  examination  shows  that  the  section  is  thicker  at  one 
side  than  at  the  other,  correct  this  by  exerting  greater  pressure  on  the 
former.  Sometimes,  when  a  slice  begins  to  break  at  one  side,  a  drop  of 
balsam,  placed  on  the  object-glass  alongside  the  slice  and  allowed  to  harden, 
may  protect  it.  Smith1  recommends  grinding  the  chip  first  in  the  form  of  a 
circular  disk.  Having  no  irregular  edge,  it  will  not  so  readily  break  away. 
In  making  the  mount  be  sure  that  no  bubbles  lie  between  the  slice  and  the 
glass,  for  above  each  bubble  the  rock  is  sure  to  break  away. 

Usually  old  emery  is  fine  enough  for  the  last  grinding,  but  Ady2  and  other 
writers  recommend  the  use  of  hone  stones.  The  writer  has  rarely  found 
this  necessary. 

If  one  is  very  finical  and  objects  to  having  the  object-slip  scratched  at 
the  four  corners,  he  may  use  as  a  protection,  four  bits  of  cover-glass  or  zinc 
cemented  on  and  afterward  removed,  as  was  suggested  by  Forbes.3  Borne- 
mann4  says  three  bits  of  glass  are  better  than  four. 

Instead  of  grinding  the  section  by  moving  the  chip  on  a  stationary  plate, 
some  form  of  grinding  machine  with  a  revolving  lap  may  be  used.  The 
procedure  is  essentially  the  same  as  that  described  for  hand  grinding  except 
that,  since  the  lap  revolves,  it  is  not  necessary  to  move  the  slide  about  so 
much.  The  section  is  first  ground  with  coarse  carborundum  and  water.  At 
the  proper  time  this  is  carefully  removed  from  both  lap  and  rock  chip, 
finer  carborundum  is  substituted,  and  the  grinding  renewed;  this  also  is 
carefully  removed  at  the  proper  time,  and  fine  emery  powder  substituted. 
Instead  of  cleaning  the  carborundum  from  the  lap  each  time,  it  is  more 
economical  in  material  and  time  to  use  three  adjacent  laps,  a  method  also 
advisable  on  account  of  the  likelihood  of  getting  a  grain  of  coarse  carborundum 
mixed  with  the  fine  in  the  first  method,  with  disastrous  results. 

In  using  a  lap  one  has  a  certain  latitude  in  rapidity  of  grinding  due  to 
the  difference  in  velocity  at  the  center  and  periphery  of  the  wheel.  In  no 
case  should  the  upper  end  of  the  spindle  project  through  the  surface  of  the 
lap,  for  by  so  doing  it  makes  useless  a  large  portion  of  the  most  useful  grinding 
surface. 

During  the  process  of  grinding,  the  chip,  whether  mounted  or  unmounted, 
should  be  so  held  between  the  thumb  and  the  first  three  fingers,  or  by  three 
fingers  alone,  that  the  finger  nails  are  not  ground  down  to  the  quick.  For 
inspection,  the  slide  may  be  removed  from  the  quickly  revolving  wheel  by 
slipping  it  toward  the  edge  and  passing  the  thumb  beneath  it.  Toward  the 
close  of  the  grinding  process  this  should  be  done  quickly  in  order  to  avoid 

1  John  Smith:    A  method  oj  making  and  mounting  transparent  rock  sections  jor  micro- 
scopic  slides.     Jour.  Postal  Microsc.  Soc.,  II  (1883),  28-33. 

2  Op.  cit. 

3  David  Forbes:  Op.  cit. 

4  See  Art.  517. 


588  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  517 

too  much  cutting  away,  although  there  is  less  danger  of  this  than  might  be 
supposed,  since  the  pressure  being  removed,  the  cutting  action  of  the  emery 
is  much  less.  If  desired,  the  final  rubbing  down  may  be  done  by  hand  on  a 
glass  plate.  As  in  hand  grinding,  experience  is  necessary,  and  numerous 
slides  will  be  spoiled  before  dexterity  is  obtained.  This  does  not  mean, 
however,  that  the  first  half  dozen  slides,  which  must  necessarily  be  spoiled, 
should  be  carelessly  ground.  Each  section,  even  from  the  beginning,  must 
be  treated  as  though  no  more  of  the  rock  material  were  to  be  obtained.1 

Surrounding  each  lap  there  should  be  a  guard  to  catch  the  water  and 
abrasive  thrown  off.  It  is  not  necessary  constantly  to  take  fresh  material, 
but  that  which  has  already  been  used  may  be  taken  up.  It  pays,  occasionally, 
to  clean  out  the  box,  wash  the  material  free  from  dust,  dry  and  sift  to  remove 
rock-fragments,  and  use  again  until  too  dull  to  cut.  It  will  be  found  that 
while  carborundum  cuts  much  more  rapidly  than  emery,  it  also  becomes  dull 
much  more  quickly,  probably  due  to  its  brittleness  and  its  consequent  more 
rapid  reduction  to  powder. 

517.  Various  Grinding  Machines. — Originally  thin  sections  were  made 
entirely  by  hand,  or  with  preliminary  grinding  on  an  ordinary  grindstone 
until  the  rock  slices  were  approximately  i  mm.  in  thickness,  after  which  they 
were  completed  by  hand.  Vogelsang2  used  a  small  emery  wheel,  constructed 
for  the  purpose,  but  suggested,  if  one  had  no  stone  at  his  disposal,  that  chips 
be  turned  over  to  a  knife  grinder  for  rough  grinding. 

One  of  the  earliest  machines  made  especially  for  section  grinding  was  that 
described  by  Sellers.3 

Another  machine,  and  one  of  a  pattern  which  has  not  been  copied  sub- 
sequently, was  that  made  by  J.  G  and  L.  G.  Bornemann.4  It  followed  the 
method  of  hand  grinding  more  closely  than  other  machines  in  that  the  grind- 
ing plate  was  stationary  and  a  contrivance  above  moved  the  mineral  chips. 
The  grinding  plate  consisted  of  an  ordinary  iron  griddle,  10  or  u  in.  in  di- 

1  Every  student  in  petrography  should  be  able  to  prepare  his  own  thin  sections.      While 
under  ordinary  circumstances  he  may  send  his  chips  away  to  be  ground,  he  may,  some  day, 
be  called  upon  to  prepare  his  own  for  immediate  use. 

Thin  sections  are  prepared,  from  material  sent  in,  by  the  following  firms: 
America, 

W.  Harold  Tomlinson,  Swarthmore,  Penn. 
Germany, 

Voigt  &  Hochgesang,  Untere  Maschstrasse  26,  Gottingen. 

F.  Krantz,  Bonn. 

2  H.  Vogelsang:    Philosophic  der  Geologic.     Bonn,  1867,  225-226. 

3  C.  Sellers:    Beschreibung  einer  Maschine  zur  Herstellung  diinner  Schliffe  von  harten 
Substanzen  fur  mikroskopische  Zwecke.     Zeitschr.  f.   gesammten  Naturwiss.,   N.   F.,  II 
(1870),  417-419. 

4  J.  G.  and  L.  G.  Bornemann  jun. :    Ueber  eine  Schleij  mas  chine  zur  Herslellung  mikro- 
skopischer  Gesteinsdiinnschliffe.     Zeitschr.  d.  deutsch.  geol.  Gesell.,  XXV  (1873),  367-373. 


ART.  517]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  589 

ameter.  It  was  placed  on  a  table  and  above  it  was  erected  a  support  for  a 
vertical  spindle,  to  the  center  of  which  was  attached  a  horizontal  pulley  and 
to  the  lower  part  a  horizontal  wooden  cross.  The  four  arms  of  this  cross 
were  pierced  with  holes  into  which  bent  wire  drags  were  placed.  The  chips, 
usually  to  the  number  of  six  or  eight,  and  of  approximately  equal  hardness, 
were  mounted  on  small  glass  plates  by  means  of  pure  beeswax,  this  material 
having  been  chosen  in  preference  to  Canada  balsam  on  account  of  its  ready 
fusibility.  If  it  was  necessary  to  grind  rocks  of  unequal  hardness  at  the 
same  time,  those  which  were  harder  were  placed  near  the  periphery  of  the 
plate  where  the  speed  of  the  drag  was  greater.  To  the  center  of  the  upper 
sides  of  the  glasses  carrying  the  chips,  posts  0.5  to  i.o  cm.  in  height  were 
cemented  with  sealing-wax,  and  over  them  were  placed  short  upright  sections 
of  close  fitting  lead  pipe.  The  latter  served  as  weights,  and  to  them  the 
drag  wires  were  attached.  As  the  chips  became  thinner,  the  lead  weights 
were  exchanged  for  others  not  so  heavy.  Usually  it  was  found  advisable 
to  allow  the  arms  of  the  cross  to  drag  the  chips  which  were  heavily  loaded 
and  push  these  which  were  not.  The  grinding  was  done  by  means  of  emery 
and  water,  the  change  from  coarse  to  fine  being  made  by  simply  changing 
the  griddle.  If  a  specimen  was  to  be  polished,  the  iron  plate  was  replaced 
by  one  of  glass  covered  with  calf-  or  buckskin  upon  which  tripoli  powder  and 
water  were  placed.  Upon  the  completion  of  the  first  face,  the  chips  were 
reversed  and  ground  on  the  other  side  until  the  slices  were  as  thin  as  "  strong 
paper."  They  were  then  mounted  on  object-slips  whose  corners  were  pro- 
tected by  three  fragments  of  cover-glasses,  not  four  as  suggested  by  Forbes, 
and  the  grinding  was  completed  with  fine  emery,  the  chip  being  frequently 
removed  and  examined. 

The  ordinary  type  of  machine  used  at  the  present  time  has  a  horizontal 
lap,  without  rock  holders,  and  is  fed  with  carborundum  or  emery  and  water. 
In  form  it  is  similar  to  all  of  the  earlier  instruments1  and  no  particular 
improvement  has  been  made.  The  table  upon  which  the  instrument  works 
should  be  of  a  height  so  that  it  does  not  tire  one's  back  or  arm  when  working, 

1  J.  Lehmann:  Einige  auf  das  Durchschneiden  von  Gesteinsstucken  und  die  Herstellung 
von  Mineral-  und  Gesteinsdunnschliffen  beziigliche  Enfahrungen.  Verb,  naturhist.  Ver. 
preuss.  Rheinl.,  Bonn.  XXXVII  (1880),  Sitzb.,  228-231. 

H.  C.  Beasley:  On  the  preparation  of  rocks  for  microscopic  examination.  Trans.  Liver- 
pool Geol.  Asso.,  Ill  (1883),  141-147. 

P.  Groth:  Physikalische  Krystallographie.    Leipzig,  1885,  667-674. 

H.  Rosenbusch:  Mikroskopische  Physiographic.     Stuttgart,  1885,  6-14. 

John  Ernest  Ady:  Op.  ciL,  1885. 

K.  J.  V.  Steenstrup:  En  formentlig  Forbedring  ved  de  saedvanlige  Slibemaskiner.  Geol. 
Foren.  i  Stockholm  Forh.,  X  (1888),  114-115. 

C.  H.  Caffyn:  A  rock-grinding  machine  for  amateurs.  Knowledge,  XXXIV  (1911), 
30-31.  Describes  a  "home  made"  cutting  and  grinding  machine  made  from  an  old 
sewing-machine  stand. 


590 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  517 


and  it  should  have  a  top  of  reasonable  size.  Williams1  used  a  table  31/2  ft. 
square  and  2  ft.  9  in.  in  height.  If  one  prefers  to  stand  while  working,  the 
table  should  be  about  40  in.  high. 


FIG.  750. — Hand  section  grinding  machine.      (Dr.  Steeg  and  Reuter.) 

Among  modern  grinding  machines  are  those  shown  in  Figs.  750  to  753. 
The  first  is  a  small  apparatus  with  a  horizontal  plate,  cast  iron  for  coarse 
grinding  and  glass  for  fine,  revolving  on  ball  bearings.  An  instrument  al- 
most exactly  like  this  is  described  by  Leiss.2 


PIG.  751. — Foot  power  lapidary's  lathe. 
I/ 1 5  natural  size.     (Fuess.) 


FIG.  752. — Motor  lap.     i/io  natural  size. 
(Fuess.) 


1  George  H.  Williams:    A  new  machine  for  cutting  and  grinding  thin  sections  of  rocks 
and  minerals.     Amer.  Jour.  Sci.,  XLV  (1893),  102-104. 

2  C.  Leiss:  Die  optischen  Instrumente.     Leipzig,  1899,  275-276. 


ART.  517]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  591 

A  foot-power  lapidary's  lathe  is  shown  in  Fig.  751.  It  consists  of  an 
iron  stand  with  a  rectangular  wooden  top  into  which  is  set  an  enameled 
iron  grinding  basin  (B).  The  horizontal  drive  wheel  (R)  is  set  in  ball 
bearings  and  is  rotated  by  double  treadles  (/  and  /i).  Grinding  disks  of 
iron,  12  to  13  cm.  (4  1/2  to  5  in.)  in  diameter,  and  a  disk  to  which  a  piece 
of  plate  glass  for  fine  grinding  may  be  attached,  are  provided  with  the 
instrument. 

Similar  to  the  enameled  basin  of  Fig.  751  is  that  shown  in  Fig.  752, 
but  the  machine  is  motor  driven,  the  belt  wheel  (ai)  being  attached  to  the 
spindle  which  operates  the  lap  and  carries  the  loose  throw-off  wheel  (a). 
The  machine  is  to  be  attached  to  the  lower  side  of  the  work  bench  by  means 




PIG.  753. — Large  automatic  grinding  machine.     1/20  natural  size.     (Fuess.) 

of  four  screws  in  the  plate  xx\,  the  basin  itself  slipping  from  above  into  a 
hole  of  proper  size.  Two  sizes  are  manufactured,  one  with  disks  of  the  same 
diameter  as  those  in  the  preceding  machine,  and  one  with  disks  of  25  cm. 
(10  in.).  The  latter  is  much  better  adapted  for  practical  work  than  the  for- 
mer. If  possible  a  bench  should  be  arranged  for  four  of  these  laps  side  by 
side,  to  avoid  the  necessity  of  continually  cleaning  up  in  changing  from  one 
grade  of  abrasive  to  another.  One  should  be  used  exclusively  for  coarse, 
one  for  medium,  and  one  for  fine  grinding.  The  fourth  could  be  used  for 
polishing  rock  faces. 

In  Fig.  753  is  shown  a  large  grinding  machine,  too  large  for  ordinary 
laboratory  purposes,  but  provided  with  an  attachment  which  might  well  be 
adapted  to  a  smaller  lap.  The  instrument  appears  to  be  intended  only 


592  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  518 

for  grinding  and  polishing  large  slabs  of  rock,  the  plate  s  for  the  attachment 
of  the  specimens  being  25  cm.  in  diameter  and  the  basin  £,35  cm.  As 
may  be  seen  from  the  illustration,  the  machine  may  be  foot  or  power  driven, 
the  motion  being  transmitted  to  the  horizontal  pulley  H  which  rotates  the 
grinding  lap.  At  the  same  time,  the  spindle  e  is  set  in  motion,  and  a  belt 
to  g  rotates  the  disk  s.  There  is  also  imparted  to  the  disk  a  forward-and- 
back  motion  by  the  eccentric  at  e,  moving  the  arm  hinged  at  c.  The  posi- 
tion of  the  plate  can  be  altered  somewhat  by  the  slide  A . 

With  such  an  attachment  to  two  laps  like  those  in  Fig.  752  and  a  plate 
(s)  approximately  4  in.  in  diameter,  to  which  four  to  six  chips  could  be 
attached  at  the  same  time,  and  used  on  a  ic-in.  grinding  disk,  the  rough  and 
intermediate  grinding  could  be  done  mechanically,  especially  if  an  automatic 
feed  for  carborundum  and  water  were  provided.  The  whole  upper  part, 

cAgs,  could  be  made  to  lift  off  and  slip 
over  the  pin  (c)  and  eccentric  (e)  of  the 
second  lap,  so  that  sections  could  be  ad- 
vanced without  removal  from  the  carrier. 
There  might  also  be  arranged  a  stop,  set 
on  screws  beneath  the  arm  A,  which 
would  prevent  too  thin  grinding  if  the 
machine  were  left  unattended  for  a  time, 
PIG.  754.— Grayson's  lap.  absolutely  preventing  the  spoiling  of  ma- 

terial by  over-grinding.     With  fairly  thin 

chips,  no  sawing  would  be  necessary,  time  being  no  great  consideration 
with  the  machine. 

The  grinding  lap  described  by  Grayson1  is  of  bronze,  10  in.  in  diameter, 
and  provided  on  the  lower  side  with  a  threaded  boss  by  which  it  is  screwed  to 
the  spindle  of  the  machine,  thus  allowing  the  whole  surface  of  the  lap  to  be 
utilized.  Tray-like  shields  or  mud  guards  of  galvanized  iron,  5  in.  deep, 
and  with  the  upper  edges  rounded  and  brass  bound,  are  provided,  although 
not  shown  in  the  illustration  (Fig.  754).  The  space  around  each  spindle  is 
raised  and  capped  so  as  to  exclude  dust  and  grit,  which  otherwise  would  soon 
ruin  the  bearings.  Somewhat  to  the  right  and  behind  the  lap  is  a  pillar 
supporting  a  horizontal  clamping  device,  and  arranged  so  that  it  may  be 
swung  by  hand  across  the  lap.  The  lower  part  of  the  supporting  rod  is  threaded 
so  that  it  may  be  raised  or  lowered  during  use.  The  machine  is  set  on  a 
table  3  ft.  2  in.  high,  and  is  driven  by  an  electric  motor  at  a  speed  of  980  revo- 
lutions per  minute. 

518.  Orienting  Devices. — For  petrographic  work  orienting  devices,  by 
which  sections  may  be  cut  at  any  desired  angle,  are  very  seldom  used. 

1  H.  J.  Grayson:  Op.  at.,  71-74. 

2  H.  Rauff :  Op.  cit. 


ART.  519]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  593 

They  are  of  great  use  in  crystallographic  work  and  reference  should  be  made 
to  the  works  mentioned  below  for  detailed  descriptions.  One  of  the  earliest 
instruments  was  that  described  by  Rauff.  In  this  the  orienting  device 
consisted  of  motions  in  two  directions  controlled  by  screws  in  the  manner 
of  a  lathe.  Another  form  is  that  found  in  the  cutting  machine  described 
by  Steinmann1  and  shown  in  Fig.  746,  and  another  is  that  by  Grayson.2 
More  accurate  is  the  device  described  by  Fuess,3  and  still  more  so  that  by 
Tutton.4  Differing  in  principle  is  Wiilfing's5  instrument,  which  is  a 
multiple-screw  device  to  be  placed  on  the  lap,  instead  of  a  goniometer  speci- 
men-holding-clamp as  are  the  others. 

519.  Mounting  the  Section. — One  of  the  operations  upon  which  the 
eventual  success  of  a  section  largely  depends  is  that  of  mounting.  Owing 
to  poor  cementation,  a  slide,  evenly  ground,  may  suddenly  break  loose 
from  the  object-glass,  or  it  may  break  away  over  a  bubble.  Sometimes  it 
may  slide  apart  in  undercooked  balsam,  or  float  apart  when  one  attempts  to 
attach  the  cover-glass. 

As  has  been  mentioned,  various  cements  are  used  to  attach  the  chip  to 
the  plate,  or  the  rock  to  the  glass.  If  the  chip  is  attached  directly  to  the 
iron  holder-plate  for  preliminary  sawing,  a  cement  of  half  beeswax  and  half 
rosin  is  strong  enough.  If  the  chips  are  first  ground  to  one  flat  surface  and 
are  then  mounted  with  Canada  balsam  on  the  object-glass,  these  plates 
may  be  attached  to  the  holder-plate  by  pure  beeswax.  This  is  one  of  the 
easiest  cements  to  remove,  since  a  slight  heating  will  loosen  the  slide  without 
melting  the  Canada  balsam  by  which  the  rock  is  attached  to  the  object  slip. 

Ady6  proposed  a  cement,  for  preliminary  mounting,  made  by  heating 
Venice  turpentine  on  a  sand  or  water-bath  and  adding  enough  orange 
shellac  to  produce,  on  cooling,  a  thoroughly  hard  yet  tough  solid.  During 
the  process  it  should  be  tested,  from  time  to  time,  by  removing  a  small 
portion  and  cooling  it.  To  attach  a  rock  slice  it  is  only  necessary  to  melt  a 
portion  of  the  cement  on  the  object-glass,  and  press  the  chip  firmly  down 
upon  it.  During  the  process  of  grinding  this  cement  does  not  take  up 

1  Gustav  Steinmann:  Op.  cit. 

2  H.  J.  Grayson:  Op.  cit.,  71. 

3  R.  Fuess:    Ueber  eine  Orientirungsvorrichtung  zum  Schneiden  und  Schleifen  von  Miner- 
alien  nach  bestimmten  Richtungen.    Zeitschr.  f.  Instrum.,  IX  (1889),  349-352. 

Also  in  Neues  Jahrb.,  1889  (II),  181-185. 

4  A.  E.  H.  Tutton:    An  instrument  for  grinding  section- plates  and  prisms  of  crystals  of 
artificial  preparations  accurately  in  the  desired  direction.     Proc.  Roy.  Soc.,  London,  LX 
(1894),  108-110.* 

Idem:  Crystallography  and  practical  crystal  measurement.     London,  1911,  681-691. 

5  E.  A.  Wiilfing:    Ueber  einen  A  p  par  at  zur  Herstellung  von  KrystallsMijfen  in  orientirter 
Lage.     Zeitschr.  f.  Kryst.,  XVII  (1889-90),  445-459. 

6  John  Ernest  Ady:    Observations  on  the  preparation  of  mineral  and  rock  sections  for  the 
microscope.    Mineralog.  Mag.,  VI  (1885),  127-133. 


594  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  519 

as  much  emery  as  does  Canada  balsam.  To  remove  the  chip  from  the  glass 
after  the  preliminary  grinding,  it  should  be  soaked  in  methylated  spirits 
for  a  few  hours.  It  should  not  be  forced  but  should  be  allowed  to  float 
off  of  its  own  accord.  It  should  then  be  transferred  to  clean  spirits  for  an 
hour  or  two  and  washed  gently  with  a  camel-hair  or  sable  brush.  This 
cement  possesses  the  advantages  of  hardening  quickly  and  of  requiring  but 
little  cooking. 

Zirkel1  recommended  16  parts  by  weight  of  thick  Canada  balsam  and  50 
parts  of  shellac.  The  shellac  should  be  dissolved  in  the  Canada  balsam  by 
heating  on  the  water-bath  for  one  or  two  hours.  As  soon  as  it  is  cool  enough, 
but  before  hardening,  it  should  be  rolled  between  the  hands  into  sticks  20 
to  30  cm.  in  length  and  i  cm.  in  diameter. 

Another  cement  suitable  for  either  preliminary  or  final  mounting  is  a 
mixture  of  equal  parts  of  gum  damar  (dissolved  in  pure  benzol)  and  Canada 
balsam.2 

Pure  Canada  balsam3  alone  may  be  used.  The  handiest  method  is  to 
evaporate  the  balsam  in  a  porcelain  dish  until  a  test  piece  is  hard  on  cooling. 
The  material  is  now  taken  up  in  balls  on  the  end  of  glass  rods  and  left  to  cool. 
To  use  the  mass  it  is  rubbed  on  a  hot  object-glass  until  the  proper  amount 
has  come  off.  Pure  paper-filtered  Canada  balsam  (Therebinthina  Canaden- 
sis)  dissolved  in  xylol  is  preferred  by  the  writer.  The  material  is  kept 
in  a  wide-mouth  bottle,  through  the  cork  of  which  a  glass  dropping  rod  is 
inserted.  Care  must  be  taken  to  keep  the  inside  of  the  bottle  neck  free 
from  balsam  or  the  cork  will  stick.  For  the  final  mounting  the  best 
grade  of  balsam  should  be  used. 

Grayson4  says  the  tenacity  and  range  of  hardness  of  the  balsam  may  be 
extended  if  a  small  quantity,  not  more  than  i  to  3  per  cent.,  of  some 
clear  and  colorless  organic  oil,  such  as  poppy,  castor,  clove,  or  linseed, 
is  added  to  it  in  the  right  proportion.  The  amount  must  be  learned  by 
experience. 

The  method  of  preparing  thin  sections  recommended  by  the  writer  is 
as  follows:  In  the  bottom  of  each  compartment  of  a  box  i  in.  deep  and 
divided  by  partitions  into  2-in.  squares,  is  placed  a  number  card  and  on  it 
the  corresponding  chip  to  be  sliced,  from  10  to  20  being  the  number  best 
handled  at  the  same  time.  The  first  chip  is  taken  and  is  ground  down  to 
a  flat  surface  on  one  side  by  the  method  described  above,  holding  the  chip  in 
the  hand  and  using  coarse,  medium,  and  fine  emery.  If  the  fragments  are 
very  thick  or  hard,  they  are  sliced  on  the  diamond  saw.  The  pieces  are  then 

1  F.  Zirkel:  Op.  cit.,  23. 

2  John  Ernest  Ady:  Op.  cit. 

3  [G.  Marpmann] :  Die modernen  Einschlusmittel.    Zeitschr.  f.  angew.  Mikrosk.,  I  (1895), 
8-1 1,  36-46.     Gives  methods  for  determining  the  kind  and  purity  of  various  embedding 
materials. 

4  H.  J.  Grayson:  Op.  cit.,  p.  76. 


ART.  519]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  595 

returned  to  the  proper  compartments  of  the  box  and  other  chips  are  ground 
until  each  has  one  flat  surface. 

It  was  formerly  customary  to  mount  the  chip  first  on  a  piece  of  thick 
glass  and  afterward  transfer  it  to  the  final  mount.  This  is  now  rarely  done, 
the  chip  being  usually  mounted  directly  upon  the  object-glass. 

Before  being  used,  object-  and  cover-glasses  should  be  made  absolutely 
clean.  For  this  purpose  a  cleaning  solution1  may  be  prepared  by  dis- 
solving 2  oz.  of  bichromate  of  potash  in  25  oz.  of  water  and  slowly  adding 
3  oz.  of  sulphuric  acid.  The  solution  should  be  left  under  a  hood  until  it  is 
cold  and  the  fumes  cease.  A  considerable  number  of  object-  or  cover- 
glasses  are  now  placed  in  a  wide-mouth  bottle  and  covered  with  the  solution. 
The  bottle  should  be  gently  tilted  a  number  of  times  to  cause  the  fluid  to 
enter  between  the  glasses  and  separate  them,  after  which  it  should  be  left 
for  three  or  four  hours.  The  solution  may  now  be  poured  back  into  the  stock 
bottle,  to  be  used  over  and  over  again,  while  the  bottle  of  glasses  should  be 
repeatedly  filled  and  emptied  with  clean 
water.  The  cover-  or  object-glasses  may 
be  left  in  the  bottle  covered  with  water  and 
taken  out  as  required  with  a  pair  of  forceps 
and  dried  with  a  linen  rag,  or  they  may  be 
placed  upright  to  dry  on  lintless  blotting 
paper.  To  support  them  on  edge,  use  may 
be  made  of  a  piece  of  wood  with  vertical  saw 
kerfs  on  the  sides,  or  two  glasses  may  be 

SO    placed    that    they    Will    mutually     Support      FIG.  755.— Cementing  oven.     (Fuess.) 

each  other. 

The  object-glasses,  having  been  cleaned,  the  proper  number  is  placed  on 
an  object-glass  heater  (Fig.  755)  whose  temperature  is  kept  between  100° 
and  150°  C.,  depending  upon  the  kind  of  balsam  mixture  used.  Grayson's 
heating  arrangement  is  a  piece  of  asbestos  or  a  metal  plate,  over  which  is 
placed  a  sheet  of  white  blotting  paper.  The  plate  is  then  laid  in  a  well- 
filled  sand-bath,  supported  by  a  tripod,  and  the  heat  of  a  Burisen  flame  so 
regulated  that  it  will  not  discolor  or  char  the  paper. 

Canada  balsam  is  next  placed  upon  the  slips.  The  writer  prefers  balsam 
dissolved  in  xylol,  using  only  a  drop  or  two — only  enough  to  squeeze  out  a 
trifle  on  all  sides  of  the  chip  when  it  is  placed  upon  it.  Too  much  balsam 
is  "messy"  and  is  likely  to  spread  over  oven,  table,  hands,  clothes,  and 
maybe  hair  as  well. 

One  may  determine  when  the  balsam  has  been  properly  cooked  by  taking 
off  a  bit  with  a  thin  glass  rod  or  burnt  match,  letting  it  cool,  and  testing  it 
on  the  finger  nail.  If  it  does  not  stick  it  is  ready  to  receive  the  chip.  Another 

1  C.  E.  Hanaman:    Note  in  Amer.  Nat.,  XII  (1878),  573-574. 

C.  Setter:  Cleaning  of  slides  and  thin  covers.     Amer.  Jour.  Microsc.,  V  (1880),  50. 


596  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  519 

method1  is  to  touch  a  bit,  while  hot,  to  the  finger,  and  draw  it  out  into  a 
fine  thread.  It  should  just  be  beginning  to  get  brittle.  The  chip  is  now 
taken  up  in  a  pair  of  forceps,  heated  for  a  moment  with  the  flat  side  up  in 
the  flame  to  drive  off  the  moisture,  and  placed  on  the  balsam.  By  first 
placing  one  edge  in  the  balsam  and  letting  the  other  go  down  gradually, 
most  of  the  bubbles  will  be  avoided.  By  pressing  down  on  the  chip  and  mov- 
ing it  about  a  trifle,  all  others  will  be  forced  out.  How  successful  one  has 
been  may  be  seen  from  the  under  side.  If  bubbles  appear,  the  slide  should 
be  reheated  and  the  bubbles  removed,  -otherwise  in  the  final  grinding,  the 
slice  is  likely  to  break  away  over  them. 

The  object-glass  and  chip  should  now  be  removed  from  the  hot  plate,  and 
the  chip  pressed  down  a  few  moments,  but  not  moved,  until  the  balsam  cools. 
After  this  the  glass  slip  should  be  marked  on  the  back,  by  means  of  a  dia- 
mond, with  its  proper  number  to  avoid  all  chance  of  confusion,  and  returned 
to  the  box.  If  numbered  on  the  face  there  is  danger  of  obliterating  the 
marks  by  grinding. 

After  having  treated  all  of  the  slides  in  this  manner,  they  are  ready  for 
the  second  face.  Chips  which  are  rather  thin  and  of  material  which  is  not 
too  hard  may  be  ground  down  without  cutting.  If  one  has  difficulty  in 
holding  the  thin  slips  without  grinding  the  finger  nails,  they  may  be  attached 
to  bits  of  plate  glass  by  means  of  beeswax  which  later  may  be  removed  by 
very  slight  heat.  If  the  chips  are  to  be  sawed,  five  or  six  are  cemented  by 
wax  and  rosin  to  the  holder-plate,  and  the  saw  is  passed  as  close  to  the  object- 
glass  as  possible.  The  slide  is  now  ground  down  to  proper  thinness,  being 
covered  with  a  drop  of  water  and  tested  under  the  microscope  from  time  to 
time.  When  the  grinding  is  finished,  the  emery-filled  balsam  is  scraped  away 
from  the  sides  of  the  chip,  and  it  is  covered  with  another  drop  of  old  or  cooked 
balsam,  which  will  require  but  little  heating,  and  a  warmed  cover-glass  is 
laid  over  the  whole.  When  thin  fresh  balsam  is  used  it  is  necessary  to  heat 
the  slide  until  the  balsam  boils.  This  softens  the  underlying  balsam,  and 
the  xylol  of  the  later  addition  creeps  under  the  chip  and  dissolves  it,  so  that, 
upon  pressing  down  the  cover-glass,  the  slice  is  likely  to  break  apart  and  be 
squeezed  out  at  the  sides.  If  only  enough  balsam  to  hold  the  cover-glass 
is  used  and  the  slide  is  set  away  for  three  or  four  days  in  a  drying  oven  main- 
tained at  a  temperature  of  45°  C.,  the  drying  will  proceed  without  danger  of 
losing  the  section. 

Instead  of  placing  the  cover-glass  upon  the  slide  immediately  after  com- 
pletion, Ady2  covered  the  finished  rock-slice  with  a  drop  of  Canada  balsam 
and  put  it  aside  for  ten  or  twelve  hours  in  a  dust-proof  box.  Another  drop 
of  balsam  was  now  added  and  a  slightly  warmed  cover-glass  placed  above  it. 

1  H.  C.  Sorby:    Preparation  of  transparent  sections  of  rocks  and  minerals.    Northern 
Microsc.,  II  (1882),  134. 

2  Op.  cit. 


ART.  519]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  597 

This  method,  however,  makes  too  thick  a  layer  of  balsam  above  the  slice  and 
causes  difficulty  when  high  power  objectives  are  used.  Bornemann1  avoided 
bubbles  under  the  cover-glass  by  placing  a  drop  of  turpentine  under  it  and 
upon  the  completed  rock-slice.  Upon  placing  a  drop  of  thin  Canada  balsam 
adjacent  to  one  side  of  the  cover,  the  balsam  rapidly  flowed  under  it,  mixing 
with  the  turpentine.  It  was  put  aside  to  harden  naturally  or  aided  by  gentle 
heat. 

Another  method  is  to  boil  the  balsam  upon  the  cover-glass  and,  when  of 
the  proper  consistency,  to  invert  it  over  the  finished  rock-slice,  previously 
warmed.  There  is  no  danger  of  the  slide  sepa- 
rating by  this  method.  Both  the  object-  and 
cover-glass  should  be  in  close  contact  with 
the  rock-slice  with  only  a  thin,  but  even,  bal- 
sam film  between.  No  instrument,2  such  as 
is  often  used  in  biologic  work  for  holding  the 

section  in  the  balsam  until  it  cools,  is  necessary,      FlG'  *'  f°r 

although  a  pair  of  tweezers  (Fig.  756) 3  which 
close  when  released,  large  enough  to  hold  a  cover-glass  transversely,  is  con- 
venient. The  cover-glass  should  be  inserted  1/16  in.  from  the  tips  of  the 
tweezers,  which  should  then  be  placed  on  the  object-glass  to  hold  it  steady 
and  be  released  when  the  cover-glass  is  in  the  proper  position.  The  drop 
of  Canada  balsam  being  convex  will  be  touched  by  the  cover-glass  first 
at  the  center,  and  as  it  is  pressed  down  the  balsam  will  be  squeezed  out 
on  all  sides.  By  so  doing,  there  is  less  danger  of  the  section  floating  away 
than  if  the  cover-glass  is  placed  down  with  one  edge  first. 

After  having  thus  mounted  and  covered  the  rock- slice,  the  excess  of  balsam 
around  the  edge  is  removed  with  a  heated  knife  blade  or  putty  knife.  The 
slide  is  then  placed  for  a  short  time  in  alcohol,  is  brushed  with  it  by  the  aid  of 
a  medium  soft  brush,  such  as  an  old  tooth  brush,  and  is  washed  in  water  and 
dried.  It  must  not  be  left  too  long  in  the  alcohol,  or  the  balsam  beneath  the 
edge  of  the  cover-glass  will  be  dissolved  out,  giving  a  projecting  edge 
which  affords  a  good  hold  for  eventually  springing  it  off  by  the  object 
clips. 

The  size  of  the  object-glasses  used  is  a  matter  of  personal  taste  and  con- 
venience. English  slides  are  usually  1X3  in.  While  they  afford  a  large 
space  at  either  end  for  labels,  they  are  too  long  for  convenience,  the  end 
projecting  over  the  stage  being  likely  to  be  struck  with  the  hand,  throwing 
the  mineral  under  examination  out  of  the  field.  The  slides  used  at  the 

1  Op.  tit.,  p.  371. 

2  L.   Henniges:    Ueber  einen  Hilfsapparat  beim  Einlegen  von  Gesteinsdiinnschli/en  in 
Kan  ad  ab  ah  am.     Centralbl.  f.  Min.,  etc.,  1911,  158-160. 

3  Dr.  Seiffert:    Eine  neue  Pincette  zum  Halten  der  Deckglaschen.     Zeitschr.  f.  angew. 
Mikrosk.,  I  (1895-6),  84. 


598  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  519 

University  of  Heidelberg  are  30X30  mm.,  a  size  too  small  to  label  with  any- 
thing more  than  the  number.  The  most  convenient  size  seems  to  be  about 
28X48  mm.  (Fig.  759).  They  are  small  enough  to  be  out  of  the  way  on  the 
stage  and  large  enough  to  label.  Those  used  by  the  U.  S.  Geological  Survey 
are  27X47  mm.  For  the  Fedorow  stage  circular  slides,  such  as  are  shown  in 
Fig.  406,  are  necessary.  For  the  new  Fuess  microscope  with  Fedorow  stage 
the  28X48  mm.  slides  may  be  used. 

It  may  sometimes  be  necessary  to  remount  an  old  rock  slice  on  account 
of  the  cracking  or  yellowing  of  the  balsam,  or  the  stripping  off  of  the  cover- 
glass.  If  the  cover-glass  is  still  in  place  and  unbroken,  a  few  drops  of  turpen- 
tine1 may  be  placed  upon  it  and  the  whole  set  on  the  heated  plate  until  the 
lower  balsam  film  melts.  The  evaporation  of  the  turpentine  will  keep  the 
upper  film  cool,  consequently  cover-glass  and  rock-slice  may  be  slipped  side- 
wise  off  the  object-glass  without  breaking.  The  object-glass  should  be 
cleaned,  or  a  new  one  taken,  and  a  few  drops  of  fresh  balsam  placed  upon  it 
and  cooked.  In  the  meantime  the  cover-glass  should  be  turned  upside  down 
and  the  old  yellowed  balsam  carefully  scraped  away  from  around  the  sides  of 
the  rock-slice.  When  the  balsam  on  the  object-glass  is  sufficiently  cooked, 
the  cover-glass  with  the  attached  section  should  be  gently  heated,  though 
not  enough  to  melt  the  balsam,  pressed  down  upon  the  object-glass,  and  the 
whole  removed  to  cool.  If  the  cover-glass  is  also  defective,  the  same  process 
is  repeated,  inverting  the  slide  and  evaporating  turpentine  on  the  bottom 
to  remove  the  cover.  If  the  cover-glass  alone  has  come  off,  scrape  away 
the  old  balsam  around  the  slice,  cook  a  few  drops  of  balsam  to  the  proper 
state  in  a  small  watch  crystal  and  pour  it  over  the  gently  warmed  old  slide 
and  cover  immediately  with  a  warmed  cover-glass. 

Sorby2  replaced  broken  object-glasses  by  removing  the  cover-glass, 
scraping  the  balsam  away  from  about  the  rock-slice,  and  covering  it  with 
plaster  of  Paris.  When  the  plaster  was  hard  the  whole  was  heated  and  the 
plaster  with  the  embedded  slice  was  pushed  off.  It  was  remounted  in  the 
usual  way.  This  method  may  be  used  to  good  advantage  with  broken  slides, 
the  pieces  being  fitted  together  and  held  with  a  piece  of  gummed  paper  on 
the  back  of  the  object-glass  before  removing  the  cover,  thus  keeping  the 
rock  fragments  in  proper  position. 

If  both  object-slip  and  cover-glass  are  broken,  the  remnants  of  the 
original  rock-slice  may  be  removed  by  placing  a  liberal  amount  of  fresh 
balsam  dissolved  in  xylol  upon  the  pieces,  and  heating  gently.  The  new 
balsam  will  work  its  way  beneath  the  slice  which  will  soon  float  upon  its 
surface.  It  may  be  transferred  to  a  fresh  object-glass,  upon  which  balsam 

1  E.  von  Fedorow:      Unhersalmethode  und  Feldspathstudien,  III.     Zeitschr.  f.  Kryst., 
XXIX  (1897-9),  617. 

2  H.  C.  Sorby:    Preparation  of  transparent  sections  of  rocks  and  minerals.    Northern 
Microsc.,  II  (1882),  137. 


ART.  520]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  599 

cooked  to  the  proper  stage  has  been  placed,  by  slightly  tilting  the  old  mount 
and  letting  the  rock-slice  float  to  its  new  position. 

SPECIAL   METHODS   FOR   PREPARING   SECTIONS    OF  UNUSUAL  MATERIAL 

520.  Friable  Material. — Soft  or  friable  material,  such  as  decomposed 
rock,  clay,  or  chalk  cannot  be  ground  in  the  ordinary  way  but  must  be  given 
a  different  treatment,  depending  upon  the  nature  of  the  material.  Forbes1 
soaked  soft  or  porous  rocks  in  turpentine,  then  in  soft  Canada  balsam,  and 
afterward  heated  them  until  quite  hard.  A  similar  method  was  used  by 
Sorby.2  Another  method  is  to  boil  the  material  in  Canada  balsam  until 
it  will  absorb  no  more  and  put  it  aside  to  harden.3  Pfaff4  prepared  chalk 
and  soft  limestone  by  inverting  the  section,  when  ready  for  the  final  grinding, 
and  rubbing  down  upon  it  with  a  very  soft  cork  and  the  finest  emery  flour. 
The  section  should  be  completely  surrounded  by  a  ring  of  Canada  balsam, 
and  if  this  breaks  away  it  should  be  replaced,  otherwise  a  few  rubs  with 
the  cork  may  break  the  edges  of  the  rock  slice.  Wichmann5  shaved  flat,  with 
a  knife,  one  side  of  soft,  fine-grained  material  and  then  rubbed  it  to  a  perfect 
plane  upon  a  dry  plate  of  glass.  The  flat  side  was  then  placed  on  an  object- 
glass  in  Canada  balsam  which  had  been  cooked  and  allowed  to  cool  to  a 
rather  viscous  state.  On  complete  cooling  the  other  side  was  shaved  down 
with  a  knife  as  much  as  possible,  cleaned  from  dust,  and  a  cover-glass  placed 
over  it  in  Canada  balsam  dissolved  in  chloroform. 

Bosscha6  prepared  sections  of  a  friable  meteorite  by  saturating  it  with 
melted  copal  gum.  He  first  ground  one  side  flat,  placed  it,  with  the  ground 
side  up,  on  a  plate  heated  to  about  125°  C.  and  upon  it  laid  pieces  of  copal 
gum,  which  melted  and  entered  the  pores.  After  cooling,  the  excess  of  gum 
was  scraped  off  and  the  section  cleaned  by  means  of  a  rag  dipped  in  ether. 

Steenstrup7  was  able  to  preserve  and  show  in  thin  sections,  by  a  double 
procedure,  the  original  arrangement  of  the  grains  in  clays.  A  piece  of  per- 
fectly dry  clay  was  ground  flat  on  one  side  upon  fine  sand  or  upon  a  glass 
plate  without  water,  and  was  then  fixed  to  a  cover-glass  by  Canada  balsam 
or  a  mixture  of  Canada  balsam  and  shellac;  the  cement  being  allowed  to 

1  David   Forbes:    On   the   preparation   of  rock   sections  for  microscopic  examination. 
Mon.  Microsc.  Jour.,  I  (1869),  240-242. 

2  H.  C.  Sorby:  Op.  cit.,  136. 

3  F.  Zirkel:    Lehrbuch  der  Petrographie,  Leipzig,  2  Aufl.,  1893,  26. 

4  F.  Pfaff:    Einiges  iiber  Kalksteine  und  Dolomite.     Sitzb.  Akad.  Wiss.,  Munchen,  XII 
(1882),  562-563. 

6  Arthur  Wichmann:  Ein  Beitrag  zur  Petrographie  des  Viti-Archipels.  T.  M.  P.  M., 
V  (1883),  33,  footnote. 

6  J.  Bosscha  Jun. :    Ueber  den  Meteorit  von  Karang-Modjo  oder  Magetan  aufJava.    Neues. 
Jahrb.,  B.B,  V  (1887),  126-144,  in  particular  127-129. 

7  K.  J.  V.  Steenstrup:    Tyndprover  af  Ler.     Geol.  Foren.  i  Stockholm  Forh.,  XII  (1890), 
647-648. 


600  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  521 

cool  somewhat  and  become  viscous  before  pressing  down  the  clay,  so  that  it 
would  not  enter  too  far  into  its  pores.  After  standing  about  twenty-four 
hours  to  permit  the  balsam  to  harden  without  heat,  the  clay  was  broken  off 
from  the  object-glass,  leaving  but  a  thin  film  upon  it.  The  fresh  face  on  the 
fragment  of  material  was  now  cemented,  without  grinding,  in  the  same 
manner  as  before,  to  another  cover-glass,  the  Canada  balsam  allowed  to 
harden,  and  the  material  again  broken  off.  If  the  clay  film  was  too  thick, 
it  was  thinned  by  gently  spraying  it  with  water  from  a  wash  bottle.  After 
drying,  Canada  balsam  and  a  cover-glass  were  placed  over  it,  and  the  cement 
allowed  to  harden  without  boiling.  The  reason  for  the  double  process  was 
that  the  mineral  particles  were  displaced  by  the  grinding,  in  the  first  flat  face, 
while  in  the  second  they  retained  their  proper  positions. 

521.  Vesicular  Rocks. — Since  the  cavities  of  pumice  and  other  vesicular 
rocks  are  closed  except  where  they  are  fractured  at  the  surface,  boiling  such 
rocks  in  balsam  is  useless.  Johnston-Lavis1  ground  such  rocks  smooth 
on  one  side,  blew  or  washed  the  dust  out  of  the  cavities,  and  placed  them 
on  a  hot  plate  to  dry.  When  well  warmed,  a  stick  of  hard  balsam  was  rubbed 
over  the  surface  and  an  abundance  of  the  cement  left  upon  in.  At  the  end 
of  a  minute  or  two  more  balsam  was  added,  if  the  first  had  sunk  in.  The 
rock  was  now  removed  from  the  hot  plate  and  allowed  to  cool  in  a  horizontal 
position,  after  which  it  was  ground  down  on  a  slab  of  sandstone,  slightly 
inclined,  over  which  a  stream  of  water  slowly  flowed.  The  rock  was  ground 
until  all  broken  septa  were  brought  flush  with  the  surface.  It  was  then 
washed,  heated,  and  more  balsam  added.  The  excess  of  balsam  was  re- 
moved by  grinding  and  the  specimen  again  washed,  after  which  it  was 
polished  by  being  rubbed  on  an  inclined  hone  upon  which  were  placed  a 
few  drops  of  a  solution  made  by  dissolving  i  pint  of  yellow  soap  in  2 
pints  methylated  spirits  and  then  adding  3  pints  of  water.  During  the 
process  of  polishing,  a  small  quantity  of  water, 'preferably  soapy,  con- 
stantly dripped  upon  the  upper  end  of  the  hone.  If  the  balsam  began  to 
"rool"  and  caused  hitching,  a  few  drops  of  the  soap  solution  were  added. 
The  specimen  was  polished  until  the  surface  was  brilliant.  It  was  then  put 
in  a  warm,  dust-free  place  to  dry,  after  which  it  was  cemented  to  a  slide  by 
hard  balsam.  The  opposite  side  was  now  ground  down  almost  to  trans- 
parency on  a  well-watered  grindstone,  then  polished  on  the  soapy  hone. 
Too  much  soap  causes  a  softening  and  saponification  of  the  balsam,  causing 
it  to  become  opaque;  too  little  causes  it  to  stick  to  the  stone  and  thus  carries 
the  section  away  with  it.  The  slide  was  finally  washed  and  dried.  When 
completely  dry  the  surface  was  brushed  with  equal  parts  of  turpentine  and 
benzol  or  chloroform  until  the  network  began  to  appear  raised.  The  slide 

1  H.  J.  Johnston-Lavis:  On  the  preparation  of  sections  of  pumice-stone  and  other  vesicular 
rocks.  Jour.  Roy.  Microsc.  Soc.,  1886,  22-24. 


ART.  5231  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  601 

was  drained,  but  not  dried,  and  balsam  dissolved  in  benzol  or  chloroform 
added,  and  the  cover-glass  placed  on  top. 

522.  Coal. — Harris1  placed  coal  for  a  considerable  time  in  turpentine, 
then  in  dilute  Canada  balsam  till  saturated.     Upon  evaporation  by  gentle 
heat,  the  balsam  gradually  hardened  and  the  coal  was  ground  down  as  any 
hard  rock. 

523.  Clays  and  Soft  Powders. — Materials,  such  as  soft  powders,  which 
do  not  need  to  be  mounted  in  such  a  way  as  to  show  the  original  texture  of 
the  rock,  may  be  mixed  to  a  paste  with  some  other  material  and  a  section 
prepared  of  the  united  mass  after  it  hardens.     Pearcey2  made  sections  of 
some  of  the  Challenger  material  by  uniting  it  with  gum  copal.     He  placed 
1/2  Ib.  of  the  best  gum  in   a  strong  glass  quart  jar  having  an  air-tight 
ground-glass  stopper,  and  added  to  it  20  oz.  of  ether  (B.  P.  sp.  gr.  0.735). 
After  standing  for  at  least  two  days,  with  frequent  shaking  or  stirring,  the 
gum  was  dissolved,  and  the  resulting  clear,  thin,  transparent  liquid  was  ready 
for  use.     The  substance  from  which  a  section  was  to  be  made  was  first  well 
dried,  then  placed  in  a  porcelain  crucible  and  twice  its  amount  of  the  gum 
copal  and  ether  poured  over  it,  care  being  taken  to  cap  the  stock  bottle  imme- 
diately.    It  was  now  placed  on  a  moderately  hot  plate,  since  the  ether  is 
very  inflammable,  and  allowed  to  simmer  until  it  had  partly  evaporated, 
when  greater  heat  was  applied.     If  the  material  was  a  fine  sand  or  ooze  it 
was  kept  well  stirred;  if  a  soft,  porous,  or  decomposed  rock,  it  was  only 
necessary  to  turn  it  several  times. 

If  the  proportions  were  right,  after  nearly  all  the  ether  had  evaporated, 
the  substance  was  of  a  stringy  nature  when  stirred.  If  it  was  found  that  too 
little  cement  remained  to  hold  the  grains  together,  more  was  added;  if  too 
much,  more  of  the  substance  as  well  as  a  little  pure  ether,  and  the  boiling 
repeated.  The  mass  was  of  a  reddish-brown  color  when  done,  and  a  small 
portion,  rapidly  cooled  by  pressing  it  against  some  cold  surface,  hardened 
immediately.  The  crucible  was  removed  and  the  material,  while  yet  warm, 
was  scraped  out  with  a  knife,  pressed  with  the  fingers  into  an  oblong  mass,  and 
molded  into  little  cylinders  about  3/4X3/4  in.  by  pressing  it  into  molds 
formed  of  strips  of  tin  tied  with  wire  and  set  on  a  piece  of  glass.  The  mass 
was  cooled  in  water,  the  mold  removed,  and  the  material  was  ready  to  cut 
like  any  other  rock  section.  If  the  sides  began  to  crumble  before  the  section 
was  thin  enough,  a  little  cement,  made  of  one  part  of  beeswax  to  four  of 
resin,  was  dropped,  while  hot,  with  a  pipette  around  the  edges  to  form  a 
support. 

1  C.  L.  Lord  and  W.  H.  Harris:  Cutting  sections  of  coal.     Science  Gossip,  1882,  136-137. 

2  F.  G.  Pearcey:    Preparing  thin  sections  of  friable  and  decomposed  rocks,  sands,  clays, 
oozes,  and  other  granulated  substances.     Proc.  Roy.  Soc.  Edinburgh,  VIII  (1884-5),  295-300. 


602  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  524 

524.  Sand  and  Other  Loose  Grains. — Sand  and  other  loose  grains  may 
be  examined  by  the  method  given  by  Thoulet,1  who  mixed  them  with  about 
ten  times  their  volume  of  zinc  oxide,  and  then  added  enough  potassium  silicate 
(water  glass)  solution  to  make  a  thick  paste.    This  paste  was  pressed  into 
sections  of  glass  tubing,  several  millimeters  in  length  and  with  parallel  ends. 
These  ends  were  covered  with  paper  and  the  material  allowed  to  dry  for  sev- 
eral days.     Sections  can  be  cut  from  such  a  mass  in  the  same  manner  as 
from  the  natural  rock. 

Mann2  mixed  the  grains  to  be  examined  with  a  paste  of  zinc  oxide  and 
phosphoric  acid  and  molded  the  mass  into  balls.  When  dry  they  were 
sectioned  as  usual. 

Retgers3  found  methods  of  embedding  grains  in  cements,  which  hardened 
later,  to  be  impracticable  for  sands,  on  account  of  the  breaking  out,  during 
the  process  of  grinding,  of  hard  minerals  such  as  zircon,  spinel,  and  corundum. 
He  crushed  the  grains  in  an  agate  mortar  to  fragments,  but  not  to  powder, 
and  immersed  them  in  a  fluid  of  high  refractive  index.  For  permanent 
mounts  he  fixed  such  grains  in  Canada  balsam. 

525.  Hydrous  Minerals. — Hydrous  minerals  cannot  be  mounted  in  the 
ordinary  way  since  they  will  lose  their  water  of  crystallization  by  the  heat 
during  the  process  of  preparation.     They  should,  therefore,  be  placed  in  the 
balsam  only  after  it  is  properly  cooked,  when  it  is  not  too  hot  and  beginning 
to  be  viscous.     The  cover-glass  should  be  laid  on  with  rather  thick  balsam  and 
set  aside  without  heating  to  harden.     Another  method  is  to  mount  them  in 
Canada  balsam  dissolved  in  ether,  and  then  place  them  to  harden,  for  several 
days,  in  a  dust-proof  box. 

526.  Minerals  Soluble  in  Water.— Minerals  soluble  in  water  should  be 
ground  with  emery  and  alcohol,  turpentine,  xylol,  etc. 

527.  The  Preparation  of  Polished  Faces  on  Rocks. — To  polish  a  rock 
which  has  been  ground  to  as  flat  a  face  as  possible  with  fine  emery,  it  is  held 
upon  a  leather-,  felt-,  or  "beaver"  cloth-covered  lap  impregnated  with  tin 
oxide  (putty  powder),  chromic  oxide,  aluminium  oxide,  or  iron  oxide  (rouge), 
and  kept  well  wet  with  water.     A  final  polish  may  be  given  by  rouge  on  a  dry 
chamois-covered  lap.     French  chalk,  rotten  stone,  or  tripoli  do  not  give  as 
good  results  as  the  oxides  mentioned  above. 

528.  Rims. — While  in  biologic  work  it  is  customary  to  surround  cover- 

1  J.  Thoulet:    Note  sur  un  nouveau  procede  d' etude  au  microscope  de  miner aux  en  grains 
tres  fins.     Bull.  Soc.  Min.  France,  II  (1879),  J88. 

2  P.  Mann:    Untersuchungen  iiber  die  chemische  Zusammensetzung  einiger  Augite  aus 
Phonolithen  und  verwandten  Gesteinen.     Neues  Jahrb.,  1884  (II),  187. 

3  J.  W.  Retgers:    Ueber  die  miner alogische  und  chemische  Zusammensetzung  der  Dunen- 
sande  Hollands  und  iiber  die  Wichtigkeit  -von  Fluss-  und  Meeressanduntersuchungen  im  All- 
gemeinen.     Neues  Jahrb.,  1895  (I),  16-74,  especially  32. 


ART.  528]  PREPARATION  OF  THIN  SECTIONS  OF  ROCKS  603 

glasses  with  a  rim  of  cement  of  some  kind,  this  is  rarely  done  with  rock  sec- 
tions, although  it  would  be  of  considerable  advantage.  It  keeps  the  air 
from  the  balsam,  thus  preventing  it  from  turning  yellow,  and  acts  as  a  guard 
to  prevent  springing  off  the  cover-glass  with  the  object-clips. 

The  rim  is  usually  put  on  with  a  brush,  the  slide  being  placed  on  a  turn 
table.  This  necessitates  the  use  of  circular  cover-glasses,  which  in  themselves 
are  advantageous,  being  less  likely  to  come  off.  If  the  cement  is  rather  thick, 
the  slide  may  be  put  on  a  turn  table  and  a  broad  band1  put  on  at  the  junc- 
tion of  cover-  and  object-glass.  A  knife  blade  may  now  be  held,  first  on 
one  side  and  then  on  the  other,  so  that  the  cement  is  heaped  up  in  a  thick 
ring.  Should  there  be  a  tendency  for  the  cement  to  run,  the  slides  may  be 
put  away  with  the  cover-glasses  downward.2 

Various  cements  have  been  used,  zinc  white3  and  asphaltum  varnish 
being  the  most  common.  The  disadvantage  of  zinc  white  is  that  it  is  too 
brittle  and  soon  breaks  away.  It  may  be  improved4  by  draining  off  the  oil 
from  the  usual  paint  and  mixing  the  latter  with  Canada  balsam  very  much 
thinned  with  chloroform.  The  mixture  should  be  of  the  consistency  of 
cream  and  flow  freely  from  the  brush.  If  it  does  not  do  so,  add  a  little 
turpentine.  The  rim  may  be  colored  as  desired  with  ordinary  artist's  oil 
paint,  and  then  varnished. 

Another  good  cement  is  dammar  varnish,  although  it  is  rather  brittle 
unless  turpentine  is  added.  It  may  be  prepared  by  mixing  gum  dammar, 
benzine,  and  turpentine  in  equal  parts,  and  setting  away  in  a  warm,  not  hot, 
place  until  dissolved.  The  clear  liquid  should  now  be  poured  off  and  allowed 
to  evaporate  until  of  the  required  consistency.  Another  method  of  prepara- 
tion5 is  as  follows:  To  4  drams  of  crushed  Indian  dammar  add  8  liquid 
drams  of  pure  benzole,  and  allow  the  resin  to  dissolve  at  the  ordinary  tem- 
perature. After  a  day  or  two  an  insoluble  residue  will  be  found  at  the  bottom 
of  the  vessel.  Carefully  decant  the  clear  liquid,  and  add  to  it  i  1/3  drams 
of  spirits  of  turpentine.  Dammar  cement  may  also  be  used  as  a  mounting 
medium  in  the  place  of  Canada  balsam.6  James7  says  a  limpid  solution  of 
dammar  may  be  obtained  by  adding  enough  benzol  to  make  a  solution  which 
is  readily  filtered  through  paper.  If  too  thin  for  immediate  use,  evaporate 

1  C.  E.  Hanaman:    Notes  on  microscopical  technology.     Amer.  Mo.  Microsc.  Jour.,  II 
(1881),  142-144. 

2  Frank  L.  James:    Microscopy.     National  Druggist,  V  (1884),  216. 

3  C.  E.  Hanaman:    Note  in  Amer.  Mo.  Microbe.  Jour.,  V  (1884),  220. 
M.  A.  Booth:  Note  in  Ibidem,  VI  (1885),  39. 

4  J.  Ford:    Dr.  Hunt's  American  cement  for  ringing  slides.     Jour.  Post.  Microsc.  Soc., 
I  (1882),  193. 

6  C.  J.  M.:    The  preparation  of  dammar  varnish  for  microscopic  purposes.     Science 
Gossip,  1882,  257. 

6Wilhelm  Pfitzner:  Die  Epidermis  der  Amphibien.  Morphol.  Jahrb.,  VI  (1880), 
footnote  479. 

7  Frank  L.  James:  Microscopy.     National  Druggist,  VII  (1885),  245. 


604  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  528 

to  the  proper  consistency.  If  the  dammar  rims  prove  too  brittle,  a  small 
amount  of  pure  rubber  dissolved  in  naphtha  may  be  added.  If  a  colored  ring 
is  desired  one  may  flow  on  a  ring  of  ordinary  water  color  before  the  varnish, 
or  the  color  may  be  mixed  with  the  latter. 

Less  brittle  than  dammar  are  rims  of  copal  varnish.  The  finest  varnish 
that  can  be  purchased  should  be  used,  Berry's  hard  finish  being  excellent, 
and  enough  dragon's  blood  may  be  added1  to  give  it  a  red  color  without 
destroying  the  transparency.  It  should  be  left  exposed  to  the  air  until  it 
becomes  rather  thick,  and  may  then  be  run  around  the  edge  of  the  cover- 
glass  in  the  same  manner  as  that  just  described.  Another  way  is  to  spin 
the  slide  on  the  turn  table  and  cut  through  the  varnish,  with  a  knife,  a  ring 
inside  and  outside  the  edge  of  the  cover-glass,  leaving  a  strip  of  the  proper 
width.  After  drying  for  a  week  the  superfluous  varnish  may  be  scraped  off. 

Another  cement2  is  composed  of  2  parts  wax  and  7  to  9  parts  of  colo- 
phony. The  latter  is  added  piece  by  piece  to  the  melted  wax  and  the 
resultant  filtered.  This  cement  is  solid  at  ordinary  temperatures  but 
readily  melts  on  being  placed  in  a  basin  of  hot  water.  It  is  insoluble  in 
water,  glycerine,  or  caustic  potash,  and,  since  it  hardens  quickly,  the  slide 
may  be  finished  at  once. 

Venice  turpentine3  is  another  substance  which  may  be  used  for  ringing 
slides.  It  may  be  prepared  by  dissolving  true  Venice  turpentine  in  enough 
alcohol  so  that  the  solution  may  be  readily  filtered.  It  is  then  placed  on 
a  sand-bath  and  evaporated  until  a  small  quantity  dropped  into  cold  water 
will  be  hard  and  break  with  a  vitreous  fracture.  Parker4  suggests  using 
square  cover-glasses.  A  piece  of  No.  10  to  12  copper  wire,  bent  into  a  right 
angle  and  having  the  short  arm  just  the  length  of  the  side  of  the  cover-glass, 
is  heated  and  dipped  into  the  prepared  turpentine,  some  of  which  adheres. 
The  wire  is  now  placed  flat  along  the  edge  of  the  cover  and  the  turpentine 
will  be  evenly  distributed  along  the  entire  side.  It  becomes  hard  immediately 
and  is  of  a  pleasing  green  tinge  from  the  copper. 

A  brown  ring  can  be  made  by  using  a  shellac  cement,5  made  by  adding 
enough  litharge  to  a  thin  shellac  varnish,  to  thicken  it.  It  should  be  applied 
in  at  least  two  coats,  the  second  added  after  the  first  is  completely  dry.  This 
cement  dries  quickly  and  becomes  dark  brown  by  exposure  to  the  air. 

1  W.:  Finishing  slides.  Amer.  Mon.  Microsc.  Jour.,  I  (1880),  123-124. 

2  Dr.  Kronig:    Einschlusskitt  fur  mikroskopische  Praparate.     Arch.  f.  Mikrosk.  Anat., 
XXVII  (1886),  657-658. 

3  Julius  Vosseler:    V ' enetianisches  Terpentin  als  Einschlussmittel  fur  Dauerprdparate. 
Zeitschr.  f.  wiss.  Mikrosk.,  VI  (1889),  292-298. 

4  C.  B.  Parker:    A  new  cement.     Amer.  Mon.  Microsc.  Jour.,  II  (1881),  229-230. 

6  Hamilton  Smith:  New  cement  and  new  mounting  medium.  Amer.  Mon.  Microsc, 
Jour.,  VI  (1885),  182. 

For  the  preparation  of  a  shellac  mounting  medium  see  Romyn  Hitchcock :  The  prepara- 
tion of  shellac  cement.  Ibide^i,  1884,  131-132. 


CHAPTER  XLH 
PETROGRAPfflC  COLLECTIONS 

FIELD  WORK 

529.  Working  Tools. — The  working  tools  of  a  field  petrologist  are  a 
geological  hammer,  a  hand  lens,  and  a  collecting-bag.  The  hammer  should 
be  made  of  the  best  cast  steel,  properly  tempered.  It  should  not  be  so  hard 
that  it  will  chip  off  at  the  corners,  nor  yet  so  soft  that  the  edges  will  round 
over.  The  form  depends  upon  the  use  to  which  it  is  to  be  put,  and  is  usually 
a  matter  of  personal  preference.  For  a  general  petrographic  hammer  one 
weighing,  without  handle,  between  a  pound  and  a  pound  and  a  quarter 
is  best.  It  should  have  at  least  one  rectangular  face,  the  other  end  being 
shaped  as  a  pick  or  wedge.  If  the  latter,  the  sharp  edge  may  run  parallel 
to  the  direction  of  the  handle  or  at  right  angles  to  it,  the  writer  preferring  the 
former  for  a  heavy  hammer  and  the  latter  for  one  which  is  light.  For  a  trim- 
ming hammer,  one  with  both  ends  rectangular  and  weighing  about  6  oz.  is  very 
convenient.  Another  useful  hammer  is  one  weighing  between  four  and  six 
pounds,  and  having  two  rectangular  faces,  each  about  1X2  1/2  in.  This  is 
especially  useful  in  breaking  off  spalls  from  a  large  block.  If  one  is  traveling 
afoot  and  but  a  single  hammer  can  be  carried,  it  should  be  of  medium  weight, 
perhaps  three-fourths  of  a  pound,  and  may  have  two  rectangular  faces,  or 
one  rectangular  and  one  wedge-shaped  with  its  edge  at  right  angles  to  the 
handle.  In  all  hammers  the  center  of  gravity  of  the  head  should  fall  at  the 
point  of  intersection  of  the  handle.  The  opening  through  it  should  be  larger 
above  than  below  so  that,  after  the  insertion  of  a  wooden  or  metal  wedge, 
there  will  be  no  danger  of  the  head  working  off.  Very  few  hammers  appear 
to  be  so  made.  The  handle  should  be  of  hickory  and  about  14  in.  in  length. 
It  should  be  trimmed  down  near  the  head  so  that  its  spring  will  absorb  all 
shock  and  not  transmit  it  to  the  hand. 

The  most  convenient  method  of  carrying  the  hammer  is  on  the  belt.  A 
case  may  be  made  of  a  circular  piece  of  leather  (Fig.  758)  about  7  in.  in  diam- 
eter, provided  on  the  front  below  the  center  with  a  horizontal  loop  for  the 
hammer  handle  and  on  the  back  with  two  that  are  vertical  for  the  belt.  The 
position  of  these  loops  should  be  such  that  when  the  hammer  is  placed  in  its 
loop  the  upper  part  of  the  leather  disk  will  fold  over  the  head  and  prevent  it 
from  slipping  out,  the  curvature  of  the  belt  around  the  body  preventing  the 
flap  from  opening.  Another  form  is  shown  in  Fig.  757-1 

1  Ferdinand  von  Richthofen:  Fiihrer  fur  Forschungsreisende.     Berlin,  1886,  14-15. 

605 


606 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  529 


For  short  excursions  the  handle  may  be  slipped  through  the  strap,  if 
such  is  provided,  at  the  back  of  the  trousers,  where  the  hammer  will  be  con- 
cealed, entirely  out  of  the  way,  and  in  no  danger  of  being  lost. 

The  method  of  carrying  the  hammer  with  its  handle  slipped  through  two 
straps  on  the  front  of  the  collecting  bag  is  not  a  good  one  since  it  is  always 
necessary  to  unstrap  the  bag  to  get  at  it. 

It  will  be  found  a  great  convenience  to  have  the  end  of  the  handle  notched 
for  4  in.  at  i-in.  intervals,  to  serve  as  a  measuring  stick  for  hand  specimens. 
The  beauty  of  a  collection  depends  largely  upon  the  uniform  size  of  the  speci- 
mens. A  thong  passed  through  a  hole  in  the  handle  and  around  the  wrist 


FIG.  757.  FIG.  758. 

FIGS.  757  and  758. — Hammer  shields. 

relieves  the  hand  from  cramping  if  one  carries  the  hammer  for  a  long  time 
while  walking.  The  hole  should  be  far  enough  up  so  that  the  handle  will 
rest  in  the  hand  when  the  thong  is  about  the  wrist.  Of  course  when  the 
hammer  is  used  the  thong  should  be  slipped  from  the  wrist. 

Hand  lenses  have  been  described  above.1 

The  collecting-bag  may  be  in  the  form  of  a  pouch  to  carry  at  the  side,  a 
knapsack,  or  a  rucksack.  If  one  is  working  afoot  the  former,  when  loaded 
with  specimens,  soon  tires  one's  shoulder.  A  knapsack  may  be  so  arranged 
with  straps  and  buckles  that  it  may  be  readily  converted  into  a  bag  to  carry 
at  the  side — if  one  objects  to  walking  through  town  with  a  bag  on  his  back.  In 
size  it  may  be  about  1 1  in.  by  1 2  in.  by  3  in.  It  should  be  divided  into  several 
compartments,  perhaps  including  one  for  map  and  note-book,  although  many 
men  prefer  to  have  the  latter  of  such  size  that  it  will  fit  the  pocket.  The  ruck- 
sack has  its  advocates  though  the  writer  finds  that  the  load  hanging  so  low  on 
the  back  is  tiresome.  Whatever  kind  of  bag  is  chosen,  it  should  be  made  of 
light  and  strong  material,  such  as  canvas;  leather  being  altogether  too  heavy. 

1  Art.  99. 


ART.  530]  PETROGRAPHIC  COLLECTIONS  607 

A  U.  S.  army  canvas  haversack,  fitted  with  straps  as  a  knapsack,  is  most 
convenient. 

530.  Hand  Specimens. — If  conditions  permit,  hand  specimens  should 
be  trimmed  to  uniform  size  and,  unless  for  special  purposes,  should  show 
fresh  faces  on  all  sides  and  no  marks  of  the  hammer.  They  should  be  about 
3  by  4  in.  in  size  and  an  inch  or  an  inch  and  a  half  thick.  The  corners  should 
be  rectangular  and  not  rounded.  If  one  is  doing  reconnaissance  work  and  his 
baggage  is  limited,  the  specimens  may  be  made  3  by  2  by  3/4  in.  or  even 
i  i/2  by  2  by  1/2  in.  It  will  be  found  much  more  difficult  to  dress  a  small, 
neat  specimen  than  a  large  one.  Each  hand  specimen  should  be  accompanied 
by  a  number  of  fresh  chips  from  the  same  piece.  These  are  to  be  used  for  thin 
sections,  and  for  possible  chemical  analysis.  For  thin  sections,  pieces  about 
i  1/2  in.  in  diameter,  free  from  cracks,  and  having  one  nearly  flat  face  should 
be  chosen.  In  wrapping  they  should  be  separated  from  the  hand  speci- 
mens by  several  sheets  of  paper.  They  may  well  be  placed  in  separate 
envelopes  and  carried  in  a  separate  compartment  of  the  collecting  bag,  for 
when  wrapped  with  the  hand  specimen  they  make  awkward  packages  which 
are  likely  to  break  open. 

Immediately  upon  collecting  the  specimen  it  should  be  labeled,  prefer- 
ably by  attaching  a  gummed  label  and  writing  the  number  upon  it  with  ink. 
The  labels  should  be  small,  3/8-in.  circular  or  oval  being  sufficiently  large. 
They  should  be  well  gummed,  better  than  ordinarily,  so  that  if  one  is  firmly 
pressed  down  into  the  irregularities  of  a  rock,  after  all  dust  has  been  blown 
off  the  latter,  and  it  is  held  down  for  a  few  moments,  there  will  be  little  danger 
of  its  coming  off.  It  is  advisable  to  place  a  locality  label  within  the  wrapper 
around  the  specimen.  This  should  be  folded  across  the  middle  to  prevent 
the  obliteration  of  the  writing.  Some  geologists  recommend  writing  the 
locality  upon  the  wrapper.  This  is  no  easier  in  the  field  than  to  prepare 
a  label,  and  it  necessitates  the  preparation  of  a  label  in  the  office  as  well. 
The  reason  for  accompanying  the  specimen  with  a  label  is  that  if  the  note- 
book, containing  the  localities  corresponding  to  the  numbers,  is  lost,  the 
specimen  will  permit  the  reconstruction  of  the  notes  to  a  certain  extent.  The 
locality  should  be  so  written  that  its  position  may  be  determined  without 
reference  to  any  points  except  such  as  are  shown  on  the  map.  That  is,  no 
label  such  as  "1/4  mi.  W.  of  camp"  should  be  used.  A  label  such  as  "300 
ft.  above  preceding  in  bed  of  creek"  is  permissible.  It  is  advisable,  every 
evening,  to  mark  the  exact  locality  of  each  specimen  by  a  number  on  a  map 
kept  for  that  purpose.  One  should  also  insert  on  the  label  information  such 
as  the  relative  position  of  a  specimen  in  a  dike,  sheet,  flow,  or  laccolith,  e.g., 
"near  top,"  "2  ft.  from  contact,"  etc.  The  labels  should  be  so  written  that 
there  will  be  no  ambiguity  in  regard  to  the  relationship  to  other  rocks. 

Specimens  should  be  collected  from  rock  in  place  and  not  from  loose 


608  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  531 

blocks,  no  matter  how  large  they  may  be,  unless  there  is  no  question  as  to 
their  source,  as,  for  example,  in  a  quarry,  talus  from  a  cliff,  etc.  This,  of 
course,  does  not  apply  to  material  collected  from  glacial  bowlders  or  terrace 
deposits. 

Certain  rocks  are  almost  impossible  to  dress  to  proper  size,  and  one  must 
do  the  best  he  can.  Thus  granite,  where  it  occurs  in  rounded  bosses,  offers 
no  chance  for  breaking  off  a  spall.  One  must  take  what  he  can  get  or  resort 
to  blasting.  The  length  of  time  necessary  to  trim  a  neat  specimen  of  a  rock 
from  which  one  can  get  a  good  spall,  should  not  be  over  two  or  three  minutes 
for  such  rocks  as  granite,  granite-porphyry,  or  limestone.  A  specimen  of 
gabbro,  pyroxenite,  or  other  tough  rock  may  take  considerably  longer. 

Where  any  variation  in  the  type  of  rock  occurs,  specimens  should  be 
collected  from  the  unusual  as  well  as  of  the  usual  phase.  This  caution  is 
hardly  necessary;  it  would  better  be  written,  collect  the  usual  as  well  as 
the  unusual.  It  not  infrequently  happens  that  upon  returning  from  the 
field  one  finds  that  the  usual  occurrences  have  been  overlooked. 

531.  Wrappers  and  Labels.— A  single  leaf  of  an  ordinary  newspaper, 
folded  in  half,  makes  a  good  wrapper.     It  should  be  so  folded  that  the  number 
of  thicknesses  on  either  side  is  as  nearly  as  possible  equal.     The  final  fold 
should  be  tucked  under  in  such  a  way  that  there  is  no  danger  of  the  wrapper 
coming  undone.     Much  more  convenient  are  specimen  envelopes  made  of 
heavy  manilla  paper.    They  should  be  at  least  8  by  10  in.  in  size  so  that 
when  an  ordinary  3  by  4  by  i  1/2  in.  hand  specimen  is  placed  in  one  corner, 
the  envelope  may  be  folded  over  first  on  one  side  and  then  on  the  other  so 
that  there  will  be  three  and  five  thicknesses  of  paper  as  a  protection  against 
rubbing.     These  bags  also  are  convenient  for  wrapping  specimens  of  tuff, 
clay,  and  so  on. 

Chips  for  thin  sections  should  be  sealed  in  small,  strong  manilla  envelopes, 
and  the  number  written  in  ink,  outside.  If  the  envelopes  are  about  21/2 
by  3  1/2  in.  in  size,  they  may  be  doubled  up  to  serve  as  a  protection  to  the 
chip  when  mailing.  There  is  usually  little  danger  of  rubbing  through  the 
wrappers,  and  it  is  unnecessary  to  use  gummed  labels  upon  chips. 

532.  Packing  Specimens  for  Shipment. — Hand   specimens   should  be 
packed  in  strong  boxes,  not  too  large,  10  in.  by  12  in.  by  14  in.,  inside  measure- 
ment, being  a  good  size.     The  wood  need  not  be  unnecessarily  thick;  a  box 
with  i  i/8-in.  ends  and  5/8-in.  sides,  wired  at  the  ends,  is  as  strong  as  one  made 
entirely  of  7/8-in.  stuff  and  not  wired.     The  wire  should  be  fairly  heavy  and 
should  be  held  in  place  by  staples  or  be  given  a  twist  around  the  heads  of 
two  or  three  nails  on  each  side  of  the  box.     The  safest  way  to  pack  hand  speci- 
mens is  to  place  a  layer  on  edge  in  the  bottom  of  the  box,  crowding  as  much 
as  possible,  and  then  fill  the  interstices  completely  with  newspaper  wads. 
A  second  and  a  third  layer  may  then  be  packed,  a  box  of  the  size  mentioned 


ART.  534] 


PETROGRAPHIC  COLLECTIONS 


609 


above  holding  three  layers  of  about  three  rows  each,  the  rows  being  rather 
irregular  on  account  of  the  lenticular  form  of  the  specimens. 

If  a  box  is  not  quite  full  the  remaining  space  should  be  crowded  with 
excelsior,  hay,  or  paper,  but  not  with  sawdust  or  other  fine  material.  The 
tighter  the  box  is  packed  the  better  it  will  stand  shipment. 


OFFICE  WORK 

533.  Accession  Catalogue. — Whether  a  collection  of  rocks  should  be 
listed  in  an  accession  catalogue  or  not  depends  upon  the  purpose  of  the  col- 
lection. For  the  ordinary  working  collection  of  material  from  one  restricted 
district  this  is  not  necessary,  but  if  the  field  embraces  a  large  territory,  or  if 
the  collection  is  that  of  an  institution,  such  a  catalogue  is  necessary.  A  very 
good  form,  following  the  plan  of  one  devised  by  Professor  Weller,  is  that 
used  at  the  University  of  Chicago.  The  pages  are  81/2  by  n  in.  in  size, 
the  entries  extending  across  two  opposite  pages  so  that  it  makes  an  avail- 
able length  of  17  in.  The  columns  are  headed  as  shown  below. 


Xo. 

Corrected 
name 

Name  under 
which  received 

Orig. 
No. 

?e^ 

Source        Locality        Remarks 

1 

- 

3 

4 

Two  lines  are  given  to  each  specimen,  every  alternate  line  being  ruled  heavier 
than  the  other.  The  numbers  on  each  left-hand  leaf  run  twice  from  o  to  9, 
permitting  twenty  entries  to  a  page.  By  beginning  each  page  with  o,  it  is 
only  necessary  to  fill  in  the  number  twice  to  a  page  instead  of  three  times. 
A  book  of  200  leaves,  giving  space  for  4000  specimens,  makes  a  convenient 
volume.  It  should  be  substantially  bound  in  canvas,  ledger  style,  in  prefer- 
ence to  leather. 

534.  Permanent  Labels  for  Hand  Specimens. — After  unpacking  speci- 
mens, they  should  at  once  be  given  permanent  numbers.  These  are  best  made 
by  painting  the  number  in  white  on  a  dark  field  in  one  corner.  The  field 
should  be  rectangular,  about  6  by  16  mm.  in  size,  and  may  be  black,  blue, 
dark  green,  or  any  other  dark  color.  Different  colors  may  be  chosen  for 
different  collections,  such  as  petrographic,  mineralogic,  or  economic.  If 
desired,  a  black  or  red  number  may  be  painted  on  a  light  field.  Enamel 
paint  seems  best  adapted  for  the  field  color.  It  should  be  rather  thick  and 

39 


610 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  535 


953 


be  flowed  on  from  a  bristle  brush  made  stiff  and  stubby  by  clamping  the  bris- 
tles close  to  the  ends  with  a  piece  of  tin.  If  the  brush  is  just  right  the  field 
may  be  made  almost  perfectly  rectangular  with  one  stroke.  After  drying, 
the  painted  area  should  be  smooth  and  glossy,  the  paint  having  been  laid  on 
thick  enough  to  fill  all  irregularities  in  the  rock.  If  the  paint  is  too  thin  it 
will  run  and  spoil  the  appearance  of  the  label.  If  it  does  not  form  a  smooth 
coat,  a  second  should  be  applied  after  the  first  is  dry.  After  a  week  or 
more  of  drying  and  when  the  paint  is  hard,  the  numbers  may  be  written 
on  with  white  paint  by  means  of  a  medium  steel  pen.  They  should  be 
neither  too  coarse  nor  too  fine,  and  as  neat  as  possible.  After  these  too  are 
dry,  a  touch  of  dammar  varnish  will  form  a  protecting  coat.  For  neatness 
and  uniformity,  the  numbers  should  always  be  placed  in  the  same  corner, 
and  as  nearly  as  possible  in  the  same  relative  position  in  all  specimens. 

535.  Labels  for  Thin  Sections. — Upon  the  object-glass  of  each  thin  sec- 
tion, a  number,  corresponding  to  the  number  on  the  hand  specimen  from 

which  it  was  taken,  should  be  scratched 
with  a  writing  diamond.  Since  this  num- 
ber is  not  easily  read,  a  paper  label  should 
be  pasted  over  it,  the  scratched  number 
being  for  safety  in  case  the  paper  label 
springs  off. 

A  convenient  way  of  numbering  slides 
so  that  the  figures  may  be   easily  read 
FIG.  759.— Labeled  thin  section.          when  the  sections  are  placed  in  boxes,  is 

shown  in  Fig.  759.     With  slides  so  num- 
bered there  is  no  excuse  for  misplacing  them  in  the  boxes  after  use. 

Northrup1  says  a  label  written  on  the  glass  with  Higgin's  water-proof 
india  ink  is  permanent  so  far  as  ordinary  treatment  is  concerned.  Before 
writing  the  label,  the  slide  must  be  made  free  from  grease  by  breathing 
upon  it  and  rubbing  with  a  dry  cloth.  Parts  of  the  label  may  be  removed, 
if  desired,  by  scratching  with  a  knife,  or  the  whole  by  rubbing  with  a  damp 
cloth. 

Besides  the  number  of  the  specimen,  the  name  of  the  rock  and  the  locality 
where  it  was  collected  may  be  written  on  the  label.  For  collections  to  be 
used  by  students,  however,  there  should  be  nothing  more  than  the  accession 
number. 

Bryan2  suggests  that  instead  of  one  thin-paper  label  at  one  end,  two  made 
of  slips  of  thick  card  be  used.  They  should  be  attached  to  the  object-glass 
at  either  side  of  the  cover.  Slides  thus  protected  may  be  placed  one  against 
another,  making  a  cabinet  unnecessary. 

1  Zae  Northrup:  A   new  method  for  labeling  microscopic  slides.     Science,  XXXVIII 
(1913),  126-127. 

2  G.  H.  Bryan:  How  to  label  microscopic  slides.     Science  Gossip,  1882,  64. 


ART.  536] 


PETROGRAPHIC  COLLECTIONS 


611 


536.  Marking  Thin  Sections. — It  is  sometimes  desirable  to  mark  a  slide 
so  that  a  certain  noteworthy  portion  may  be  readily  found  on  a  future  occa- 
sion. One  of  the  most  convenient  instruments  for  this  purpose  is  the  object 
marker1  shown  in  Fig.  760.  The  mineral,  whose  position  is  to  be  marked,  is 
placed  in  the  center  of  the  field  under  the  cross-hairs,  after  which  the  objective 
is  removed  and  the  object  marker  substituted.  The  diamond  D  is  controlled 
by  the  screw  b  and  the  spring  F,  and  may  be  placed  out  of  center  as  far  as  the 
slider  a  will  permit.  5  is  a  weak  spiral  spring  by  means  of  which  the  inner 
cylinder  C  is  pressed  downward  in  the  casing  H.  It  is  kept  from  falling  out 
by  the  screw  c  which  works  in  a  slot.  Upon  depressing  the  tube,  the  diamond 


FIG.  760. — Section  marker.     Natural 
size.     (Fuess). 


FIG.  761. — Section  marker.      (Reichert.) 


touches  the  cover-glass  with  greater  or  less  pressure  depending  upon  the 
amount  of  the  depression.  If,  now,  the  stage  of  the  microscope  be  rotated, 
the  diamond  will  scratch  a  circle  upon  the  glass,  its  size  depending  upon  the 
amount  of  the  displacement.  In  the  form  shown  in  Fig.  761  the  sizes  of  the 
circles  are  shown  by  the  graduations  on  the  screw  Sr. 

Instead  of  a  permanent  scratch,  one  may  desire  to  place  upon  the  cover- 
glass  a  mark  to  indicate  temporarily  a  certain  portion,  as  for  example  for 
micro-photography.  For  this  purpose  there  may  be  used  a  holder  similar 
to  the  above  but  provided  at  the  lower  end  with  a  metal  ring  which,  when 
inked  by  a  stamping  pad  and  depressed  until  it  touches  the  cover-glass, 
leaves  a  circular  mark.  A  spring  prevents  any  injury  to  the  slide.  Cones 
with  different-sized  rings,  interchangeable  with  the  first,  are  furnished  with 

1  C.  Leiss:  Die  optischen  Instrumente,  etc.  Leipzig,  1899,  248-249. 

See  also  P.  Schiefferdecker:  Ueber  einen  Apparat  zum  Markirenvon  Theilen  mikrosko- 
pischen  Objecte.  Zeitschr.  f.  wiss.  Mikrosk.,  Ill  (1886),  461-464. 

R.  Fuess:  Apparat  zur  dauernden  Kennzeichnung  bemerkenswerther  stellen  in  mikro- 
skopischen  Objecten  oder  Praparaten.  Neues  Jahrb.,  1895  (I),  280-281. 


612 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  537 


the  device.  *  If  the  circular  ring  were  of  rubber  there  would  be  less  danger 
of  breaking  the  slide  and  a  better  ring  would  be  impressed  upon  the  glass. 

With  a  mechanical  stage  provided  with  guide  strips,  any  desired  mineral 
may  be  found  on  a  subsequent  occasion  if  both  vernier  readings  are  noted 
and  the  slide  is  inserted  in  the  same  position  as  it  was  before.  It  is  necessary, 
however,  to  use  the  same  microscope  for  the  determination.  The  position 
of  any  point  may  likewise  be  determined  with  the  Hirschwald  stage  as  modi- 
fied by  Johannsen.2 

537.  Cases  for  Thin  Sections. — The  manner  of  storing  sections  depends 
upon  the  size  of  the  collection.  If  the  number  of  specimens  is  few  they  may 
be  kept  in  boxes  such  as  are  shown  in  Fig.  762.  The  septa  shown  in  Fig.  763 


FIG.  762. — Box  for  thin  sections.     (Dr.  Steeg  and  Reuter.) 

are  of  compressed  paper  and  are  of  sufficient  length  so  that  slides  a  few  milli- 
meters shorter  or  longer  than  normal  may  be  inserted  without  difficulty. 
Being  of  paper  they  are  much  thinner  than  would  be  necessary  were  they  saw 
kerfs  in  wood  strips,  consequently  many  more  sections  may  be  placed  in  a 
box  of  a  given  size. 


FIG.  763. — Septa  in  section  box.     (Dr.  Steeg  and  Reuter.) 

For  a  larger  collection  of  sections  a  neat  cabinet  may  be  made  by  fastening 
a  number  of  boxes  like  Fig.  762  tightly  together  in  a  frame  like  a  sectional 
book-case,  using  small  brass  ring  fasteners  as  drawer  pulls. 

1  Manufactured  by  Klonne  und  Mtiller,  Berlin.     Originally  described  by  P.  Francotte, 
Bull.  Soc.  Beige  de  Micr.,  XI  (1882),  48.  *     Reviewed  in  Jour.  Roy.  Microsc.  Soc.,  V  (1885) : 
325- 

2  See  Art.  109,  supra. 


ART.  538] 


PETROGRAPHIC  COLLECTIONS 


613 


If  it  is  desired  to  keep  the  sections  flat,  cases  such  as  shown  in  Fig.  764 
may  be  used.  Such  cases  take  up  more  space,  are  more  expensive  per  unit , 
and  the  sections  are  more  easily  disarranged  than  in  those  previously  described, 
but  they  permit  the  entire  label  to  be  read  without  moving  the  slide.  If 
sections  are  labeled  as  suggested  above  this  is  hardly  necessary,  and  only  in 
such  regions,  as  in  the  southwestern  states,  where  the  summer  heat  is  so  great 
that  the  balsam  softens  and  the  section  slides  away,  is  it  necessary  to  keep 
sections  flat. 

A  third  type  of  case,  intermediate  between  that  for  vertical  and  that  for 
flat- lying  sections,  is  one  in  which  the  sections  are  slipped  into  inclined  grooves. 


FIG.  764. — Slide  cabinet.     (Bausch  and  Lomb.) 

There  seems  to  be  no  particular  advantage  in  this  method  so  far  as  cheapness 
or  saving  of  space  is  concerned. 

Merrill1  made  cheap  cases  for  storing  thin  sections  by  folding  manilla 
wrapping  paper  into  pleats.  The  slides  were  placed  on  end  between  the 
folds,  which  acted  as  springs,  the  whole  being  placed  in  proper-sized  boxes. 

538.  Card  Catalogue. — The  working  petrologist  should  make  a  card  cat- 
alogue of  all  specimens  collected.  This  method  of  keeping  a  record  pos- 
sesses several  advantages  over  any  other  method.  Cards  for  the  same  area 
or  of  the  same  type  of  rock  may  be  studied  together,  and  the  cards  may  be 
arranged  or  rearranged  in  any  manner  most  convenient  for  the  time  being. 
This  is  of  special  importance  in  cataloguing  transition  types  which  may  have 
to  be  transferred  from  one  group  to  another.  The  method  also  is  elastic, 
and  the  descriptions  may  be  extended  at  will.  The  form  of  card  is  a  matter  of 
personal  preference.  All  that  is  necessary  is  that  all  possible  information  be 
written  upon  it.  The  aim  should  be  to  write  the  descriptions  so  that  another 

1  George  P.  Merrill:  A  cheap  form  of  box  for  microscopic  slides.  Science,  XX  (1892), 
298-299. 


614 


MANUAL  OF  PETROGRAPHIC  METHODS 


[ART.  538 


person,  from  the  description  alone,  may  obtain  a  mental  picture  of  the  rock. 
The  form  given  below  for  systematic  collections  is  also  very  good  for  field 
collections. 

The  following  description  of  the  card  catalogue  of  the  University  of 
Chicago  collection,  which  is  a  modification  of  that  used  by  the  U.  S.  Geo- 
logical Survey1  for  its  reference  collection,  may  serve  as  a  model  for  other 
institutions. 

The  rock  specimens  are  arranged  in  the  order  of  accession,  and  the  thin 
sections,  which  bear  corresponding  numbers,  are  numerically  arranged  in  a 
case  such  as  was  suggested  above,  that  is,  in  a  series  of  boxes  similar  to  Fig. 
762  arranged  in  tightly  fitting  cases  holding  2000  slides  each,  and  occupying  a 
space  of  12  by  26  by  7  in. 

The  regular  descriptive  cards  are  4  by  6  in.  in  size,  and  are  similar  to  that 
shown  below.  They  are  arranged  in  numerical  order  in  the  files,  the  number 
at  the  left  on  the  card  being  the  accession  number. 


826 
Leucite  basanite. 


Lava  of  1760,  Vesuvius,  Italy. 


Megascopic 
Microscopic 


Medium   gray.     Very  many   stout   augite   prisms,  dark  green  in  color. 
Groundmass,  dark  gray,  aphanitic. 
Texture. — Porphyritic,  nearly  sempatic. 

Phenocrysts. — About  45  per  cent.     Megaphyric. 

Short,  stout  prisms  and  regular  basal  sections. 
Groundmass. — Holocrystalline,  hypautomorphic. 
Constituents. 

Phenocrysts. — Augite  90  per  cent.,  leucite  5  per  cent.,  olivine, 

5  per  cent. 

Groundmass. — Leucite  40  per  cent.,  augite  25  per  cent.,  plagio- 
clase  20  per  cent.,  magnetite  8  per  cent.,  olivine  5  per  cent, 
biotite  2  per  cent. 
A  ccessory. — Apatite. 
Secondary. — Chlorite. 

Noteworthy. — Leucite  with  inclusions.     Poikilitic  augite. 
Remarks.     — Section  rather  thick.     Specimen  taken  from  near  surface. 
Occurrence. — Lava  flow  of  1760. 
Literature.  — Rosenbusch:  Mikroskopische  Physiographic,  II,  1907,    1376- 

1379- 
Zirkel:  Lehrbuch,  III,  1894,  13. 


The  form  of  the  card  differs  for  granular  and  for  porphyritic  rocks  in  that 
the  constituents  of  the  former  are  arranged  under  the  headings:  essential, 
minor  accessories,  occasional  accessories,  and  secondary.  Rocks  which  have 
been  analyzed  have  their  analyses  written  on  the  margin  at  the  left. 

To  make  the  collection  available  for  all  purposes,  several  series  of  index 
cards  are  provided.  The  first  is  a  classification  index.  This  is  divided  into 

1  Published  with  the  permission  of  the  Director,  U.  S.  Geological  Survey. 


ART.  538]  PETROGRAPHIC  COLLECTIONS  615 

families  and  subfamilies,  and  under  each  division  is  a  card  giving  the  numbers 
and  localities  of  all  rocks  of  this  kind  in  the  collection.  This  classification  is 
temporary  and  can  readily  be  changed.  In  the  U.  S.  Geological  Survey 
collection,  the  descriptive  cards  are  themselves  arranged  according  to  rock 
terms.  For  the  use  of  students  it  has  seemed  better  to  arrange  these  cards 
numerically,  similar  rocks  being  readily  found  from  the  cross  reference  clas- 
sification index. 

The  advantage  of  keeping  the  descriptive  cards  in  numerical  order  is  that 
one  is  not  tied  down  to  any  system  of  classification.  The  petrographical 
index  may  be  arranged  to  suit  any  system,  or  several  indices  may  be  made  for 
several  different  systems.  If  a  rearrangement  is  desired,  it  is  only  necessary 
to  revise  comparatively  few  cards  and  not  the  complete  collection. 

As  a  specimen  of  the  classification,  the  subdivisions  of  the  granite  family 
used  in  the  temporary  arrangement  at  the  University  of  Chicago  are  given  in 
part  below. 

TOO  Normal  alkali-lime  series. 

no  Predominating  feldspar  orthoclase. 
in  Granite-rhyolite  family. 
1 1 1 .  i  Plutonic. 

1 1 1 . 1 1  Leucocratic. 

1 1 1 . 1 1 1  Alaskite. 

1 1 1. 1 12  etc. 
in .  12  Normal. 

1 1 1 . 1 2 1  Biotite  granite  =  granitite. 
in  .122  Two  mica  granite, 
in .  123  Amphibole  granite. 

111.1231  Hornblende  granite. 

in .  1232  etc. 
in .  124  Pyroxene  granite. 

1 1 1 . 1 241  Augite  granite. 

111.1242  Diopside  granite. 

111.1243  etc. 
111.125  Topaz  granite, 
in .  126  Garnet  granite. 
111.127  etc. 

111.13  Melanocratic. 

111.131  Melano-granite. 
1 1 1 .  2  Hypabyssal. 

1 1 1 .  2 1  Leucocratic. 

1 1 1 .  2 1 1  Alaskite  porphyry, 
in .  22  Normal. 

1 1 1 .  2  2 1  Granite  porphyry. 

1 1 1 . 3  Effusive. 

111.31  Leucocratic. 
111.311  Tordrillite. 

111.32  Normal. 

111.321  Rhyolite. 

111.322  Rhyolite  porphyry. 

§uartz  porphyry, 
rthoclase  porphyry,  etc. 

1 1 1. 4  Differentiation  rocks. 

111.41  Leucocratic. 

111.411  Aplite. 

111.412  Pegmatite. 

111.413  etc. 

111.42  Melanocratic. 
111.421  etc.,  etc. 


616  MANUAL  OF  PETROGRAPHIC  METHODS  [ART.  538 

Under  these  subdivisions  are  arranged  cards  such  as  the  following: 


GRANITE.     Biotite  granite  (Granitite). 


7 
ii 

21 

22 

36 

38 

63 

69 

84 

101 

I03 

104 
106 

112 

US 
117 


Gross  Bieberau,  Odenwald,  Germany. 

Muhltal,  Eberstadt,  Odenwald.     (Hornblende  bearing). 

Heidelberg,  Baden. 

Burkersdorf,  Erzgebirge,  Sachsen  (Porphyritic). 

Eibenstock,  Erzgebirge,  Sachsen. 

Konigshain,  Schlesien. 

Grasstein,  Tirol. 

Frauenthal,  Bohemia,  Austria. 

Koritnicza,  Hungary. 

Baveno,  Lago-Maggiore,  Italy  (Red). 

Bovey  Tracey,  Devonshire,  England. 

High  Downs,  Cornwall,  England. 

Dalbeattie,  Scotland. 

Ross  of  Mull,  Scotland  (Porphyritic). 

Nystad,  Finland  (Rapakiwi). 

Wyborg,  Finland  (Rapakiwi). 


Besides  this  index  there  are:  an  index  of  noteworthy  minerals;  an  index  of 
other  noteworthy  features,  such  as  textures,  etc.;  a  geographical  index. 
The  following  is  an  example  of  the  first: 


APATITE. 


809 
816 

848 

859 
864 
867 
869 

873 
899 

937 

954 


Medium  size  in  granite.     Fine. 

Usual  small  laths,  showing  parting,  in  granite. 

Irregular  grains  and  laths  in  granite. 

In  syenite. 

In  syenite. 

Large  and  small  grains,  in  syenite. 

Small  laths  in  syenite. 

Cross  parting  well  shown,  in  quartz  monzonite. 

Small,  in  diorite. 

Showing  inclusions  and  corrosion,  in  andesite. 

Showing  many  inclusions  and  corrosion,  in  andesite. 


Etc.  etc. 


The  geographical  index,  including  cross  reference  cards,  is  subdivided, 
at  the  present  time,  as  follows:  (Further  subdivisions  may  be  added  as 
needed.) 

Ascension  Islands. 
Argentina. 
Austria-Hungary. 
Austria. 

Bohemia. 

Erzgebirge. 
Mittelgebirge. 
Bukowina. 
Dalmatia. 

Carinthia  (Karnten). 
Carniola  (Krain). 
Coastland. 
Erzgebirge,  see  Bohemia,  Erzgebirge. 


ART.  538]  PETROGRAPHIC  COLLECT  IOXS  617 

Galicia. 

Lower  Austria. 

Moravia  (Mahren). 

Salzburg.         , 

Silesia  (Schlesien). 

Styria  (Steiermark). 

Tyrol  and  Vorarlberg. 

Upper  Austria. 
Bosnia. 
Bolivia. 
Brazil. 
Hungary. 

Croatia  and  Slavonia. 

Hungary  (Ungarn). 

Transylvania  (Siebenburgen). 
Belgium. 
Canada. 

British  Columbia. 
Ontaria. 
Quebec. 
ChiU. 
Egypt. 
England. 
France. 

Miscellaneous. 

Yosges  (Vogesen),  see  Germany,  Alsace-Lorraine. 
Germany. 

Alsace-Lorraine  (Elsass-Lothringen) . 
Baden. 

Odenwald,  see  Hessen,  Odenwald. 
Bavaria  (Bayern). 

Fichtelgebirge. 

Pfalz. 

Rhon  Gebiet,  see  Thuringen  States,  Rhon  Gebiet. 

Spessart  Gebiet. 
Brunswick. 

Harz  Gebiet. 

Erzgebirge,  see  Austria-Hungary,  Austria,  Bohemia,  Erzgebirge. 
Eifel,  see  Prussia,  Rhine,  Eifel. 
Fichtelgebirge,  see-  Bavaria,  Fichtelgebirge. 
Harz  Gebiet,  see  Brunswick,  Harz  Gebiet, 
Hessen. 

Odenwald. 
Odenwald,  see  Hessen,  Odenwald. 

Pfalz,  see  Bavaria,  Pfalz. 
Prussia. 

Brandenburg. 

East  Prussia. 

Eifel,  see  Prussia,  Rhine,  Eifel. 

Hanover. 

Hessen-Nassau . 

Rhon  Gebiet,  see  Thuringen  States,  Rhon  Gebiet. 
Spessart  Gebiet,  see  Bavaria,  Spessart  Gebiet. 

Pomerania  (Pommern). 

Posen. 

Rhine  (Rheinland,  Rhenish  Prussia). 
Eifel. 
Siebengebiige. 

Saxony. 

Harz,  see  Germany,  Brunswick,  Harz. 

Schleswig-Holstein. 

Silesia  (Sch^sien). 

Westphalia. 

West  Prussia. 


618  MANUAL  OF  PETROGRAPHIC  METHODS  "[ART.  538 

Rhenish  Bavaria,  see  Bavaria,  Pfalz. 

Rhon  Gebeit,  see  Thuringen  States,  Rhon  Gebiet. 

Saxony. 

Erzgebirge,  see  Austria-Hungary,  Austria,  Bohemia,  Erzgebirge. 

Schwarzwald,  see  Baden. 

Siebengebirge,  see  Prussia,  Rhine. 

Spessart  Gebiet,  see  Bavaria,  Spessart  Gebiet. 

Thuringen  States. 
Rhon  Gebiet. 

Vogesen,  see  Alsace-Lorraine. 

Wiirtemberg. 
Italy. 
Ireland. 
Norway. 
Mexico. 
Peru. 
Portugal. 
Russia. 

Finland. 

Great  Russia,  Archangel. 

Northern  Caucasia. 

Trans-Caucasia. 

Ural  Mountains. 
Scotland. 

^  Island  of  Skye. 
Spain. 
Sweden. 
Switzerland. 
United  States. 

Arizona. 

Arkansas. 

CaUfornia,  etc.,  etc.,  etc. 
Venezuela. 
Wales. 


APPENDIX 


Letter 

Name 

Corresponding  letter  in 
English 

A 

ex. 

Alpha  .  .             

A 

B 

r 

ft 

Beta  
Gamma  

B 
G 

A 

8 

Delta 

D 

E 

£ 

Epsilon 

Short  E 

Z 

Zeta   . 

Z 

H 

rj 

Eta  .  . 

Long  E 

e 

6  d 

Theta  .  . 

Th 

I 

K 

K 

Iota  
Kappa  

I 
K 

A 

A 

Lambda  

L 

M 

Mu 

M 

N 

Nu  

N 

0 
II 
P 

0 

7T 
p 

Xi  
Omicron  
Pi    
Rho 

X 

Short  O 
P 
R 

2 

a  s 

Sigma 

s 

T 

r 

Tau 

T 

T 

Upsilon   .  .   . 

u 

<P  <t> 

Phi.  .  . 

F 

X 

X 

Chi  
Psi 

Ch 
Ps 

12 

CO 

Omega  

Long  O 

i 

USEFUL  FORMULA 

TRIGONOMETRIC 


(i)  sin^=-     (Fig.  765) 


(2)  cos  A  =- 


(3)  tan 

(4)  cot 

(5)  sec 

(6)  esc 


FIG.  765. 


619 


620  MANUAL  OF  PETROGRAPHIC  METHODS 

Each  of  the  six  principal  functions  may  be  expressed  in  terms  of  the  other  five  as  follows : 


sin 

cos 
tan 

cot 
sec 

CSC 

A/I  —  cos2    (7) 

tan       (8) 

1           (a) 

1          (ii) 

x/sec2—  T 

Vi+tan2 
1          d~) 

\/i+cot2 
cot 

(IO) 
1          fiO 

CSC 

\/l—  sin2  (12) 
-7==   (17) 

V  esc2      i  cT6-) 

x/i+tan2 

V  i+cot2 

i         ,     . 

sec 

9  /•          \ 

CSC 

i 

Vi-cos2 

VI  —sin2 

COS 
COS 

(2-*) 

l^T    (24) 

cot      (I9) 

sec2  —  i   (20) 

i 

Vcsc2-i(2l) 

\/i  —  sin2  /     s 

sm 

I 

VI  —  cos,2 

I       (^ 

x/sec2-i     ^2S; 

sec 
'      (if)} 

Vcsc2—  i  (26) 

CSC 

X/I+COt2   (^Q) 

,  —      -    (27) 
V  i  -sin2 

-s-  <"> 

cos         (28) 
1             (33) 

V-^F<«> 

cot 

VCSC2-!     (3I) 

\/i+cot2  (35) 

VI  -cos2 

\/sec*-i      (36) 

(37)  sin2 


(39) 
(40) 
(41) 
(42) 
(43) 
(44) 
(45) 
(46) 
(47) 
(48) 
(49) 
(50) 
(51) 
(52) 
(53) 
(54) 


cos  ^4 


sin  (  —  A  )  =  —  sin  A 
cos  (  —  A)=cos  A 
tan  (-4)  =  -tan  ^ 
cot  (-^)  =  -cot  A 
sec  (-4)=sec  A 
esc  (  —  A)  =  —csc  A 
sin  (90°  +  A)  =  cos  A 
cos  (90°  +  ^!)  =  —sin  A 
tan  (90°  +  ^!)  =  -cot  A 
cot  (9o°-M)  =  -tanyl 
sec  (90°  +  A}  =  -esc  A 
esc  (9o°+yl)=sec/l 
sin  (a+/3)  =  sin  a  cos  /3  +  cos  a  sin  ft 
cos  (a+j8)  =cos  a  cos  /3  —  sin  a  sin  ft 
,     ,     ,      tan  a  +  tan  /3 


N 
(55)  cot  (a+ft)  = 


i  —  tan  a  tan  /? 
cot  a  cot  /3  — 


(56) 
(57) 

/  0 
(58) 


cot  /3  +  cot  a 

sin  (a  —(3)  =  sin  a  cos  ft  —  cos  a  sin  ft 
cos  (a—  /3)  =  cos  a  cos  0-f  sin  a  sin  0 

tan  a  —  tan  j8 
tan  («-/3)=- 

i  +  tan  a  tan  0 

cot  a  cot  /3+  i 


(60)  sin  «+sin/3  =  2  sin  i/2(a+j8)  cos  i/2(a-/3) 

(61)  cos  a  +  cos/3  =  2  cos  i/a(a+j8)  cos  i/2(a—  j9) 

(62)  sin  a-sin/3  =  2  cos  i/2(a+/3)  sin  1/2(01—  ft) 

(63)  cos  a-cos  |8=  —  2  sin  1/2  (a+0)  cos  i/  2(01—  ft) 
sin_a_+sin   /5_tan_i/2_(a^|-/S) 

4)  sin  a  -sin  0~tan~i/2(a^"j8) 

(65)  sin  2e*  =  2  sin  a  cos  a 


APPENDIX 


621 


(65)  cos  2«  =  cos2  a  — sin2  a 

(67)  cos  2a  =  i  — 2  sin2  a 

(68)  cos  2a  =  2  cos2  a— i 

2  tan  a 
(60)  tan  20;  = 

i  —  tan2  a 

cot2  a  —  i 
(70)  cot2«  =  — 


i  —  cos  a 
(78)    USEFUL  VALUES  OP  NATURAL  TRIGONOMETRIC  FUNCTIONS 


Angle 

Sin                  Cos 

Tan 

Cot 

Sec 

Csc 

0° 

o 

I 

O                              oo 

I                               oo 

30° 
45° 

| 

iVi 

Jv/J 

I 

IVS 

2 

\/2 

60° 

iv7 

1 

N/i 

•J\/  -2 

2 

IVj 

93° 

I 

o 

OO 

o 

00 

I 

1  80° 

o 

—  I 

o 

oo 

—  I 

oo 

270° 

—  I 

o 

00 

0 

oo 

—  I 

360° 

0 

I 

O 

00 

1 

oo 

CARTESIAN  GEOMETRIC 

Rectilinear  equation  to  a  right  line. 

y  =  m'x+b  (Fig.  766). 
(79)  DP  =  m'OD+OF 


Rectangular  equation  to  a  right  line. 

co  =  90°  (Fig.  767). 

(80)  y  =  mx+b,  where  w  =  tan  6. 

Polar  equation  to  a  right  line. 

(81)  p  cos  (<f>  —  a)=p,  where  p  =  perpendicular  to  OC  (Fig.  767). 


622 


MANUAL  OF  PETROGRAPHIC  METHODS 


Equation  to  the  circle,  the  origin  being  at  the  center. 

(82)  x*+y*  =  R*  (Fig.  768). 

Axial  equation  to  the  ellipse. 

(83)  bW+a*y*  =  aW(Fig.  769) 

or    2+2-=Ij  where  CD  =  x,PD  =  y,  CA=a,  CB=b. 


FIG.  767. 


Equation  of  the  hyperbola  referred  to  its  axes. 

(84)  bW-a*y*  =  a*b* 
where  CD  =  x,  DP  =  y,  CA=a,  CF  =  c,  c*  = 

Equation  to  the  equilateral  hyperbola. 

a  =  b  in  equation  (103)  and 

(85)  x2  —  yz  —  az  (Fig.  770). 


FIG.  768. 


(Fig.  .770). 


FIG.  770. 


Equation  of  the  tangent  to  the  ellipse. 

(86)  ?r+flr-x- 

Equation  of  the  tangent  to  the  circle,  origin  at  the  center 

(87) 

Equation  of  the  tangent  to  the  hyperbola. 

(88)  ^T-^J 
Equation  of  the  normal  to  the  ellipse. 

(89)  £_  9L 


APPENDIX  623 


Equation  of  the  normal  to  the  circle. 

(90)  ?-y 

Equation  of  the  normal  to  the  hyperbola. 
(9.)  + 


CONVERSION  TABLES  FOR  WEIGHTS  AND  MEASURES 
LINEAR  MEASURE 

i  millimeter  =    0.0394    inch. 
10  mm.   =  i  centimeter  =    0.3937  inch. 
10  cm.     =  i  decimeter   =    3.937    inches. 
10  dcm.  =  i  meter  (m)  =  39.37      inches. 

i  inch    =  25.399  rnm. 
i  foot     =    0.30479+  m. 
i  yard   =    0.91439+  m. 

Paris  line    =  2.  2558  mm.  =  0.089  m- 

12  Paris  lines      =  i  Paris  inch  =  27.07  mm. 

12  Paris  inches  =  i  Paris  foot   =  0.3248  m. 

6  Paris  feet       =  i  toise            =  i .  9490  m. 

English  duodecimal  line  =     2. 1166  mm. 
English  inch  =  25 . 3997  mm. 

Prussian  line    =  2 . 1802  mm. 
Prussian  foot    =  0.31385  m. 

Vienna  line    =     2. 1952  mm. 
Vienna  inch  =  26.3419  mm. 


MEASURES  OF  CAPACITY 

Cubic  meas.      Dry  measure         U.  S.  liquid  measure 

i  liter  (i)  =  1000  c.c.  =     0.908  quarts  =     1.0567  quarts. 
61.02201.  in.        33. 8  ounces. 


WEIGHTS 

Amount  of  water  at  max-     Avoirdupois  weight 
imum    density    to    which 
equal 

i  milligram  (mg.)         =  i  cubic  millimeter  =     0.0154  grain. 

1000  mg.  =  igram(grm.)  =    i  cubic  centimeter  (c.c.)  =  15.432  grains. 

1000  grm.  =  i  kilogram  (kg.)          =  1000  c.c.  =  i  liter  =    2 .  2046  pounds. 


624  MANUAL  OF  PETROGRAPHIC  METHODS 

USEFUL  RECIPES  l 

Acid-proof  cement  for  glass  cells,  etc. 

(a)  Resin  24  parts,  red  ochre  4  parts,  calcined  plaster  of  Paris  2  parts, 
linseed  oil  i  part.     Unite  by  stirring  together  when  melted.* 

(b)  Shellac  in  alcohol. 

(c)  Asphaltum  in  turpentine. 

Water-proof  cement. 

(a)  Shellac  4  parts,  borax  i  part.     Boil  in  a  little  water  until  dissolved. 
To  use  heat  till  pasty.* 

(b)  Dissolve  as  much  gutta-percha  as  possible  in  10  parts  carbon  bi- 
sulphide and  i  part  turpentine.* 

(c)  Melt  shellac,  and  mold  into  sticks.     Warm  the  articles  to  be  cemented 
sufficiently  to  melt  the  shellac  when  applied. 

(d)  For  cementing  glass,  repairing  troughs,  etc     Dissolve  5  to  10  parts 
gelatine  in  100  parts  water;  add  zoper  cent,  saturated  bichromate  of  potassium 
solution;  mix  thoroughly  and  keep  in  a  dark  place.     After  using  the  cement 
the  articles  are  exposed  to  sunlight,  by  the  action  of  which  the  medium  is 
rendered  insoluble  in  water.     (M.  I.  Cross:  Knowledge,  XXVI  (1903),  285- 
286.) 

Cement  for  mending  rock  specimens. 

(a)  Dissolve  shellac  in  alcohol.     Apply  to  both  parts  and  bind  together  till 
dry. 

(b)  Equal  parts  red  and  white  lead  mixed  with  boiled  linseed  oil  to  a 
proper  consistency.     Color  is  objectionable. 

(c)  White  cement.     Plaster  of  Paris  in  a  saturated  solution  of  alum.* 

(d)  White  cement.     Melt  together  resin'8  parts  and  wax  i  part,  then  stir 
in  plaster  of  Paris  4  parts.     Heat  pieces  to  be  mended.* 

(e)  Plaster  of  Paris  in  a  solution  of  gum  arabic  to  which  a  few  drops  of  oil 
of  cloves  have  been  added.     Color  suitably  with  a  small  amount  of  lamp 
black,  umber,  or  ochre. 

(f)  Gray  cement.    Litharge  20  parts,  dry  lime  i  part.     Make  into  putty 
with  linseed  oil.     Sets  in  a  few  hours.* 

Cement  for  attaching  rock  chips  to  holder  plate. 

Besides  those  described  in  the  text,  the  following  may  be  used: 
Black  resin  4  parts,  beeswax  i  part.     Melt  and  add  i  part  whiting  pre- 
viously heated  red  hot  and   still  warm.     The  proportions  may  be  varied 
within  a  wide  range. 

1  Recipes  starred  have  not  been  tested  by  the  writer.    They  are  given  for  convenience 
without  recommendation. 


APPENDIX  625 

Cement  for  attaching  leather,  felt,  etc.,  to  metal. 

(a)  Gelatine  dissolved  in  acetic  acid. 

(b)  Common  glue  56  parts  by  weight.     Add  3  1/2  parts  gum  arable. 
Stir  to  an  even  paste  with  water  over  fire.     Remove  and  add  slowly  31/2 
parts  nitric  acid.  * 

Cement  for  attaching  glass  laps  to  metal  holders. 

(a)  Shellac  i  Ib.  dissolved  in  methylated  spirits  i  pint. 

(b)  Fine  litharge  2  parts,  white  lead  i  part.     Make  into  a  paste  with  3 
pints  boiled  linseed  oil  and  i  of  copal  varnish.     Add  more  litharge  and  lead 
if  required.  * 

Glue  for  attaching  labels  to  metal  or  glass. 

(a)  Add  a  little  calcium  chloride  to  the  glue.     It  will  prevent  cracking.  * 

(b)  Break  yellow  glue  into  small  pieces,  soak  in  cold  water  for  a  few  hours, 
then  pour  off  the  water.     Place  the  softened  glue  in  a  wide-mouthed  bottle 
and  add  enough  glacial  acetic  acid  to  cover.     The  glue  will  dissolve  more  read- 
ily if  placed  on  the  water-bath. 

(c)  Dextrine  mucilage.     Dissolve  2  oz.  dextrine  in  i  oz.  acetic  acid  diluted 
with  5  oz.  water.     When  dissolved  add   i   oz.    alcohol.     (Microsc.   Bull., 
II  (1885),  46.) 

(d)  Dissolve  120  grm.  gum  arabic  in  1/4  liter  of  water  and  30  grm.  of 
gum  tragacanth  in  a  similar  quantity.     After  a  few  hours  shake  the  traga- 
canth  solution  until  it  froths  and  then  add  the  gum  arabic  solution.     Strain 
through  linen  and  add  150  grm.  glycerine,  previously  mixed  with  2  1/2  grm. 
oil  of  thyme.*     (Zeitschr.  f.  angew.  Mikrosk  II  (1896),  151.) 

Ink  for  writing  on  glass. 

(a)  Water  glass  (sodium  silicate)  i  to  2  parts,  fluid  Chinese  white  i  part.* 
(Zeitschr.  f.  angew.  Mikrosk.,  I  (1895),  l83-) 

(b)  With  the  rubber  stopper  of  a  hydrofluoric  acid  bottle  touch  the  slide 
or  bottle  where  the  label  is  desired.     A  frosted  surface  will  result  upon  which 
the  label  may  be  written  with  a  lead  pencil.     Such  marks  will  withstand 
steam  and  ordinary  handling,  and  may  be  removed  with  a  rubber  eraser  when 
desired.     This  is  especially  useful  for  labels  on  beakers,  flasks,  etc.,  used  in 
analysis.     (Science,  XXXVII  (1913),  561-562.) 

Simple  formula  for  mixing  any  grade  of  alcohol  required. 

Let  P  represent  the  grade  per  cent,  of  the  alcohol  on  hand,  P'  the  grade 
per  cent,  required,  v  the  number  of  volumes  of  water  to  be  added  to  one  volume 
of  P  to  make  alcohol  Pr,  and  x  the  number  of  volumes  of  P  desired  to  change 

to  P'.    Then  ~~=Pr1  and  v=^£-.     (Ohio  Naturalist,  VI  (1906),  352- 
353-) 

40 


626. 


MANUAL  OF  PETROGRAPHIC  METHODS 


NATURAL  SINES  AND  COSINES 


A 

Sin 

Cos 

A 

Sin 

Cos 

A 

Sin 

Cos 

• 

0° 

0  .  000000 

0.002909 
0.005818 
0.008727 
0.011635 
0.014544 

I  .  OOOO 

90° 

30' 
40' 
So' 

0.1305 
0.  1334 
0.1363 

0.9914 
0.9911 
0.9907 

30' 

20' 
10' 

15° 

0.2588 

0.9659 

75° 

10' 

20' 
30' 
40' 

So' 

I.  0000 

I  .0000 

I.  0000 
0.9999 
0.9999 

50' 
40' 
30' 

2C 
10' 

10' 
20' 
30' 
40' 
So' 

0.2616 
o.  2644 
0.2672 
o.  2700 
o.  2728 

0.9652 
0.9644 
0.9636 
0.9628 
0.9621 

So' 
40' 
30' 
20' 

10' 

8° 

0.  1392 

0.9903 

82° 

10' 

20' 
30' 

4C' 

So' 

o.  1421 
o.  1449 
0.1478 
o.  1507 
0.1536 

0.9899 
0.9894 
0.9890 
0.9886 
0.9881 

So' 
40' 
30' 
20' 
10' 

1° 

0.017452 

0.9998 

89° 

1  6° 

0.2756 

0.9613 

74° 

10'  0.02036 
20'  0.02327 
3o'jo.  02618 
4o'io.  02908 
50'  0.03  199 

0.9998 

0.9997 
0.9997 
0  .  9996 
0.9995 

50' 

40' 
30' 
20' 

10' 

10' 
20' 
30' 

40' 
SO' 

0.2784 
o.  2812 
o.  2840 
o.  2868 
0.2896 

0.9605 
0.9596 
0.9588 
0.9580 
0.9572 

50' 

40' 
30' 
20' 

10' 

9° 

o.  1564 

0.9877 

81° 

10' 

20' 
30' 
40' 

So' 

0.  1593 

o.  1622 
o.  1650 
o.  1679 
o.  1708 

0.9872 
0.9868 
0.9863 
0.9858 
0.9853 

So' 
40' 
30' 

20' 
10' 

a° 

0.03490 

0  .  9994 

88° 

17° 

0.2924 

0.9563 

73° 

10' 

20' 
30' 
40' 
So' 

0.03781 
0.04071 
0.04362 
0.04653 

0.04943 

0-9993 
0.9992  - 
0-9990 
0.9989 
0.9988 

So' 
40' 
30' 
20' 
10' 

10' 
20' 
30' 

4°' 
SO' 

0.2952 
0.2979 
0.3007 
0.3035 
0.3062 

0.9555 
0.9546 
0.9537 
0.9528 
0.9520 

50' 
40' 
30' 
20' 
10' 

10° 

0.1736 

0.9848 

80° 

10' 

20' 
30' 
40' 
50' 

o.  1765 
0.1794 

0.1822 
o.  1851 
0.1880 

0.9843 
0.9838 
0.9833 
0.9827 
0.9822 

So' 
40' 

30' 

2O'| 

10' 

3° 

0.05234 

0.9986 

87° 

1  8° 

0.3090 

0.95H 

72° 

10' 

20' 
30' 

40' 

So' 
4° 

0.05524 
0.05814 
0.06105 
0.06395 
0.06685 

0.9985 
0.9983 
0.9981 
0.9980 
0.9978 

So' 
40; 
30' 
20' 
10' 

10' 

20' 
30' 
40' 
50' 

0.3118 

0.3145 
0.3173 
0.3201 
0.3228 

0.9502 
0.9492 
0.9483 
0.9474 
0.9465 

SO' 
40'. 
30' 
20' 
10' 

11° 

0.1908 

0.9816 

79° 

10' 

20' 

30' 

40' 

So' 

0.1937 
0.1965 
0.1994 

0.2022 

o.  2051 

0.9811 
0.9805 
0.9799 
0.9793 
0.9787 

So' 
40' 
30' 
20' 
10' 

0.06976 

0.9976 

86° 

19° 

0.3256 

0.9455 

7i° 

10' 
20' 
30' 
40' 
So' 

0.07266 
0.07556 
0.07846 
0.08136 
0.08426 

0.9974 
0.9971 
0.9969 
0.9967 
o  .  9964 

So' 
40' 
30' 
20 
10' 

10' 
20' 
30' 
40' 
So' 

0.3283 
0.3311 
0.3338 
0.3365 

0.3393 

0.9446 
0.9436 
0.9426 
0.9417 
0.9407 

50' 
40' 
30' 
20 
10' 

12° 

o.  2079 

0.9781 

78° 

10' 
2O' 
30' 

40' 

So' 

0.2108 
o.  2136 
o.  2164 

0.2193 
O.  2221 

0.9775 
0.9769 
0.9763 
0.9757 
0.9750 

50' 
40' 
30' 

20' 

10' 

5° 

10' 

20' 
30' 
40' 
So' 

0.08716 

0.09005 
0.09295 
0.09585 

0.09874 

o.  10164 

0.9962  85° 

20° 

0.3420 

0.9397 

70° 

0.9959 
0.9957 
0-9954 
0.9951 
0.9948 

So' 
40' 
3o' 

20' 

10' 

10' 

20' 
30' 
40' 

SG' 

0.3448 
0.3475 
0.3502 
0.3529 
0.3557 

0.9387 
0-9377 
0.9367 
0.9356 
0.9346 

SO' 

40' 
30' 
20' 

10' 

13° 

o.  2250 

0.9744 

77° 

10' 
20' 
3C' 
40' 
So' 

o.  2278 

0.2306 
0.2334 
0.2363 
0.2391 

0-9737 
0.9730 
0.9724 
0.9717 
0.9710 

So' 

% 

2O'i 

10' 

6° 

0.10453 

0  .  9945 

84° 

21° 

0.3584 

0.9336 

69° 

10' 
20' 
30' 
40' 

So' 

o.  10742 
o.  11031 
o.  11320 
o.  11609 
o.  11898 

0.9942 
0.9939 
0.9936 
0.9932 
0.9929 

So' 
40' 
30' 
20' 
10' 

10' 

20' 
30' 
40' 
50' 

0.3611 
0.3638 
0.3665 
0.3692 
0.3719 

0.9325 
0.9315 
0.9304 
0.9293 
0.9283 

So' 
40' 
30' 
20' 
10' 

14° 

O.24I9 

0.9703 

76° 

10' 

20' 

30' 

40' 

So' 

0.2447 
0.2476 
0.2504 
0.2532 

o.  2560 

o  .  9696 
0.9689 
0.9681 
0.9674 
0.9667 

So'1 
40', 
30' 
20' 

lO'j 

7° 

o.  12187 

0.992S 

83° 

22° 

0.3746 

0.9272 

68° 

10' 

20' 
30' 

o.  12476 
o.  12764 

0.13053 

0.9922 
0.9918 
0.9914 

So' 
40' 
30' 

10' 
2O' 
30' 

0.3773 
o.  3800 
0.3827 

0.9261 
0.9250 
0.9239 

50' 
40' 
30' 

15° 

0.2588 

0.9659 

75° 

Cos 

Sin 

A 

Cos 

Sin 

A 

Cos 

Sin       A 

APPENDIX 


627 


NATURAL  SIXES  AND  COSINES.— Continued 


A 

Sin 

Cos 

A 

Sin 

Cos 

A 

Sin 

Cos 

30' 
40' 
So' 

0.3827 
0.3854 
0.3881 

0.9239 
0.9228 
0.9216 

30' 
20' 

10' 

30° 

0.5000 

0.8660 

*,' 

30'  0.6088 
4o'j  o.6m 
50'j  0.6134 

0.7934 
0.79i6 
0.7898 

30' 
20' 
10' 

10' 
20' 

30' 
40' 
50' 

0.5025 
0.5050 
0.5075 
o.  5100 
0.5125 

0.8646 
0.8631 
0.8616 
0.8601 
0.8587 

so' 

40' 
30' 
2O' 

10' 

59°~~ 

So' 

40; 
30' 
20' 

10' 

23° 

o  .  3907 

0.9205  67° 

38° 

0.6157  0.7880  52° 

10' 
20' 
30' 
40' 

50' 

0.3934 
0.3961 
0.3987 
0.4014 
0.4041 

0.9194 
0.9182 
0.9171 
0.9159 
0.9147 

so; 
40' 

30' 
20' 
10' 

10' 
20' 
30' 
40' 
50' 

0.6180 
0.6202 
0.6225 
0.6248 
0.6271 

0.7862 
0.7844 
0.7826 
0.7808 
0.7790 

So' 
40' 
30' 
20' 
10' 

3i° 

0.51501  0.8572 

10' 
20' 
30' 
40' 
V  So' 

0.5175 
0.5200 
0.5225 
0.5250 
0.5275 

0.8557 
0.8542 
0.8526 
0.8511 
o  .  8496 

24° 

0.4067 

0.9135 

66° 

39° 

0.6293 

0.7771  51° 

10' 

20' 

30' 
40' 

SO' 
^5° 

0  .  4094 
0.4120 
0.4147 
0.4173 
0.4200 

0.9124 
0.9112 
0.9100 
0.9088 
0-9075 

So' 
40' 

30' 

20' 
10' 

10' 
20' 
30' 
40' 
50' 

0.6316 
0.6338 
0.6361 
0.6383 
o  .  6406 

0.7753 
0-7735 
0.77i6 
0.7698 
0.7679 

50' 
40' 
30' 
20 

10' 

32° 

0.5299 

0.8480 

58° 

10' 
20' 
30' 

40' 
So' 

0.5324 
0.5348 
0.5373 
0.5398 
0.5422 

0.8465 
0.8450 
0.8434 
0.8418 
o  .  8403 

So' 
—  40' 
30' 
20 
10' 

0.4226 

0.9063 

65° 

40° 

0.6428;  0.7660 

50° 

10' 
20' 

30' 

40' 

So' 
26° 

0.4253 
0.4279 
0.4305 
0.4331 
0.4358 

0.9051 
0.9038 
0.9026 
0.9013 
0.9001 

So' 

40; 
30' 
20' 
10' 

10' 
20' 
30' 
40' 
50' 

0.6450 
0.6472 
0.6494 
0.6517 
0.6539 

0.7642 
0.7623 
0.7604 
0.7585 
0.7566 

0.7547 

So' 
40' 
30' 
20 
10' 

33° 

0.5446 

0.8387 

57° 

10' 
20' 
30; 

40' 
50' 

34°" 

0.5471 
0.5495 
O.S5I9 
0-5544 
O.S568 

0.5592 

0.8371 
0.8355 
0.8339 
0.8323 
0.8307 

So' 

40' 
30; 

20' 
10' 

56°~ 

0.4384 

0.8988 

64° 

41° 

10' 

20' 
30' 
40' 
SO' 

0.6561 

49° 

10' 

20' 

30' 
40' 
50' 

0.4410 
0.4436 
0.4462 
0.4488 
0-4514 

0.8975 
0.8962 
o  .  8949 
0.8936 
0.8923 

So' 

40' 
30' 
20 
10' 

0.6583 
0.6604 
0.6626 
0.6648 
0.6670 

0.7528 
0.7509 
0.7490 
0.7470 
0-7451 

50' 
40' 
30; 

20' 
10' 

0.8290 

10' 
20' 
30' 
40' 
So' 

0.5616 
o.  5640 
0.5664 
0.5688 
0.5712 

0.8274 
0.8258 
o.  8241 
0.8225 
0.8208 

50' 

40' 
30' 

20' 

10' 

27°   j  0.4540 

0.8910  63° 

142° 

0.6691 

0.7431 

48° 

10' 
20' 
30' 

40' 
So' 

0.4566 
0-4592 
0.4617 
o  .  4643 
o  .  4669 

0.8897 
0.8884 

o.  8870 
0.8857 
0.8843 

50' 
40' 
30' 
20' 
10' 

10' 
20' 
30' 
40' 

50' 

0.6713 
0.6734 
0.6756 
0.6777 
0.6799 

0.7412 
0.7392 
0-7373 
0-7353 
0-7333 

50' 

40' 
30' 

20' 
10' 

35° 

0.5736 

0.8192 

55° 

10' 
20' 
30' 
40' 
50' 

0.5760 
0.5783 
0.5807 
0.5831 
0.5854 

0.8175 
0.8158 
0.8141 
0.8124 
0.8107 

SO' 

40' 
30' 

20' 
10' 

28° 

10' 
20' 

30' 
40' 
So' 

0.4695 

0.8829 

62° 

43° 

0.6820 

0-7314 

47° 

0.4720 
0.4746 
0.4772 
0.4797 
0.4823 

0.8816 
0.8802 
o.  8788 
0.8774 
0.8760 

50' 
40; 

30' 
20 
10' 

10' 

20' 

30' 

40' 
50' 

0.6841 
0.6862 
0.6884 
0.6905 
0.6926 

0.7294 
0.7274 
0.7254 
0.7234 
0.7214 

So' 
40' 
30' 
20' 
10' 

46° 

;36° 

0.5878  0.8090  54° 

10' 

20' 

30; 
40' 
50' 

37° 

0.5901 
0.5925 
0.5948 
0.5972 
0-5995 

0.8073 
0.8056 
0.8039 
0.8021 
0.8004 

50' 
40; 
30 
20' 

10' 

29° 

0.4848 

o  .  8746 

61° 

44° 

o  .  6947 

0.7193 

10' 

20' 
30' 
40' 

So' 

0.4874 
0.4899 
0.4924 
0.4950 

0.4975 

0.8732 
0.8718 
0.8704 
0.8689 
0.8675 

50' 
40' 
30' 
20' 
10' 

10' 

20' 

30' 

4°' 

50' 

0.6967 
0.6988 
0.7009 
o.  7030 
0.7050 

0.7173 
0-7IS3 
0.7133 
0.7112 
0.7092 

0.7071 

50' 
40' 
30; 

20' 
10' 

45°~ 

0.6018 

0.7986 

53° 

10' 

20' 
30' 

0.6041 
0.6065 
0.6088 

0.7969 
0.7951 

0.7934 

SO' 
40' 
30' 

30°    1  0.5000 

o  .  8660 

60° 

45° 

0.7071 

Cos 

Sin 

A 

Cos 

Sin 

A 

Cos 

Sin 

A 

628 


MANUAL  OF  PETROGRAPHIC  METHODS 


NATURAL  TANGENTS  AND  COTANGENTS 


A 

Tan 

Cot 

A 

Tan 

Cot 

A 

Tan 

Cot 

0° 

o.ooooco 

00 

90° 

30' 

4°' 
50' 

o.  1317 
o.  1346 
o.  1370 

7-5958 

7-4287 
7.2687 

30' 
20' 
10' 

15° 

0.2679 

3-7321 

75° 

10'  0.002909 
20'  0.005818 
30'  0.008727 
40'  0.011636 
50'  0.014545 

343-7737 
171-8854 
114.5887 
85.9398 
68.7501 

So' 
40'! 
30'! 

20' 

10' 

10' 
20' 
30' 
40' 
50' 

o.  2711 
0.2742 
0.27-73 
o.  2805 
0.2836 

3.6891 
3-6470 
3  •  6050 
3-5656 
3-5261 

So' 
40' 
30' 
20' 
10' 

8° 

o.  1405 

7-II54 

82° 

10' 
20' 
30' 
40' 
So' 

0.1435 

o.  1465 
o.  1495 

0.1524 

0.  1554 

6.9682 
6.8269 
6.6912 
6.5606 
6.4348 

So' 
40' 
30' 
20' 
10' 

81° 

i°     jo.  017455  57-290089° 

1  6° 

o.  2867 

3-4874 

74° 

10' 
20' 
30' 
40' 
So' 

0.02036 
o.  02328 
0.02619 
0.02910 
0.03201 

49-  1039 
42.9641 
38.1885 
34.3678 
31-2416 

28.6363 

50' 
40' 
30' 
20' 
1C' 

88° 

10' 

20' 
30' 
40' 
50' 

o.  2899 

0.2931 
o.  2962 
0.2994 
0.3026 

3  -  4495 
3.4124 
3-3759 
3-3402 
3.3052 

50' 
40' 
30' 
20' 
10' 

9° 

0.1584 

6.3138 

10' 
20' 
30' 
40' 
So' 

o.  1614 
o.  1644 
o.  1673 
o.  1703 
0.  1733 

6.  1970 

6.0844 
5-9758 
5.8708 
5  •  7694 

So' 
40' 
30' 
20' 

10' 

2° 

0.03492 

17° 

0.3057 

3-27C9 

73° 

10'  0.03783 
20'  0.04075 
30'  0.04366 
40'  0.04658 
50'  0.04949 

3°     0.05241 

26.4316 
24.5418 
22.9038 
21.4704 
20.  2056 

So' 
40'  ; 
30' 
20' 
10' 

10' 
20' 
30' 

40' 
50' 

0.3089 
0.3121 

0.3153 
0.3185 
0.3217 

3.2371 
3-2041 
3.  1716 
3-1397 
3-  1084 

50' 
40' 
30' 
20 
10' 

10° 

0.1763 

5.6713 

80° 

10' 

20' 

30' 

40' 
50' 

0.1793 
o.  1823 
0.1853 
o.  1883 
o.  1914 

5.5764 
5  •  4845 
5-3955 
5  .  3093 

5-2257 

So' 
40' 
30' 

20 

10' 

19.0811 

87° 

18° 

0.3249 

3-0777 

72° 

10' 
20' 
30' 
40' 
So' 

0.05533 
0.05824 
o.  06116 
0.06408 
0.06700 

18.0750 
17-  1693 
16.3499 
15.6048 
14.9244 

50' 
40' 
30' 
20 

10' 

10' 
20' 
30' 

4°' 
So' 

0.3281 
0.3314 

o  .  3346 
0.3378 
0.3411 

3-0475 
3-0178 
2.9887 
2.9600 
2.9319 

So' 
40' 
30' 
20 
10' 

7*~ 

11° 

o.  1944 

5.1446 

79C 

10' 
20' 

30' 
40' 

50' 

o.  1974 
o.  2004 
0.2035 
0.2065 
0.2095 

5.0658 
4.9894 
4.9152 
4-8430 
4.7729 

So' 
40' 
30' 
20' 
10' 

4° 

o  .  06993 

14-3007 

86° 

19° 

0.3443 

2.9042 

10' 

20' 
30' 

40' 
So' 

0.07285 
0.07578 
0.07870 
0.08163 
0.08456 

13.7267 
13-1969 

12.  7O62 
12.2505 
11.8262 

So' 
40' 
3o'j 

20'[ 

10' 

10' 
20' 
30' 
40' 
50' 

0.3476 
0.3508 
0.3541 

0.3574 
0.3607 

2.8770 
2.8502 
2.8239 
2.7980 
2.7725 

so; 

40 
30' 
20' 

10' 

70° 

12° 

o.  2126 

4.7046 

78° 

10' 
2O' 
30' 
40' 

50' 

0.2156 
0.2186 

0.2217 
0.2247 

o.  2278 

4.6382 
4.5736 
4-5107 
4  •  4494 
4.3897 

So' 
40' 
30' 
20' 
10' 

5° 

10' 
20' 
30' 
40' 
So' 

0.08749 

11.4301  85° 

20° 

0.3640 

2.7475 

0.09042 
0.09335 
0.09629 
0.0992.3 
o.  10216 

II.OS94 

10.  7119 
10.3854 
10.0780 
9.7882 

So' 

40'! 

3o'| 

20' 

10' 

10' 
2C' 

30' 

40' 
50' 

0.3673 
0.3706 
0.3739 
0.3772 
0.3805 

2.7228 
2.6985 
2.6746 
2.6511 
2.6279 

so; 

40 
30 
20' 
10' 

13° 

0.2309 

4-3315 

77° 

10' 
20' 
30' 
40' 
5c' 

0.2339 
0.2370 
o.  2401 
0.2432 
0.2462 

4-2747 
4-2193 
4-  1653 
4.  1126 
4.0611 

So' 
40' 
30' 
20' 
10' 

6° 

o.  10510 

9.5144 

84° 

21° 

0.3839 

2.6051 

69° 

10' 

20' 
30' 
40' 

So' 

o.  10805 
o.  i  1099 
O.H394 
o.  11688 
o.  11983 

9.2553 
9.0098 
8.7769 

8.5555 
8.3450 

So' 
40'; 
30' 
20' 
10' 

10' 

20' 

30' 

40' 

So' 

0.3872 
0.3906 
0.3939 
0.3973 
0.4006 

2.5826 

2.  5605 
2.5386 
2.5172 
2.4960 

so; 
40' 
30; 
20' 

10' 

14° 

0.2493 

4.0108 

76° 

10' 

20' 
30' 
40' 

50' 

0.2524 
0.2555 
0.2586 
0.2617 
o  .  2648 

2.9617 
3.9136 
3-8667 
3.8208 
3.7760 

So' 
40'  i 
30' 

20'! 

10' 
75~~ 

7° 

o.  12278 

8.144383° 

22° 

o  .  4040 

2.4751 

68° 

10' 

20' 
3O' 

0.  12574 
o.  12869 
o.  13165 

7-9530 
7-7704 
7.5958 

So' 
40' 
30' 

A 

10' 

2O' 
30' 

0.4074 
0.4108 
0.4142 

2.4545 
2.4342 
2.4142 

so; 
40' 
30' 

15° 

o.  2679 

3.7321 

Cot 

Tan 

Cot     Tan 

A 

Cot 

Tan 

A 

APPENDIX 


629 


NATURAL  TANGENTS  AND  COTANGENTS.— Continued 


A 

Tan 

Cot 

A 

Tan 

Cot 

A 

Tan 

Cot 

30' 
40' 
50' 

0.4142 
0.4176 
0.4210 

2.4142 
2-3945 
2.3750 

I 
30' 
20' 
10' 

30° 

0.5774 

1.7321 

60° 

30' 
40' 

50' 

0.7673 
0.7720 
0.7766 

1.3032 
1-2954 
1.2876 

30' 
20' 
10' 

10' 

20' 
30' 
40' 
50' 

0.5812 
0.5851 
o.  5890 
0.5930 
0.5969 

1.7205 
1.7090 
1.6977 
i  .  6864 
1.6753 

50' 
4°' 

£ 

10' 

23° 

0.4245 

2.3559  67° 

38° 

0.7813 

1-2799  52° 

10' 
20' 
30' 

40' 

50' 
24° 

0.4279 
0.4314 
0.4348 
0.4383 
0.4417 

2.3369 
2.3183 
2.2998 
2.2817 
2.2637 

50' 
40; 
30' 
20' 
,0' 

10' 
20' 
30' 
40' 

50' 

o.  7860 
0.7907 
0.7954 
o.  8002 
0.8050 

1.2723 
1.2647 
1.2572 
I  -  2497 
1.2423 

So' 
40' 
30' 

20' 

10' 

31° 

o  .  6009 

•  1-6643  59° 

10' 

20' 
30' 
40' 

So' 

0.6048 
0.6088 
0.6128 
0.6168 
0.6208 

J-6534 
i  .6426 
1.6319 
1.6212 
I  .6107 

50' 
40' 
30' 

20' 

10' 

0.4452 

2  .  2460 

~-\ 

50' 
40' 
30' 
20' 
10' 

39°    )  0.8098 

1-2349 

Si° 

10' 

20' 
30; 

40' 
SO' 

0.4487 
0.4522 
0.4557 
0.4592 
0.4628 

2.2286 
2.2113 
2.  1943 
2.  1775 
2.  1609 

10' 
20' 
30' 
40' 
50' 

0.8146 
0.8195 
0.8243 
0.8292 
0.8342 

1.2276 
1.2203 
I.  2131 
1.2059 

1.1988 

So' 
40' 
30' 

20' 

10' 

32° 

o  .  6249 

I  .  6003 

58° 

10' 

20' 

30' 

40' 

So' 

0.6289 
0.6330 
0.6371 
0.6412 
0.6453 

1-5900 
1-5798 
1.5697 
1-5597 
1-5497 

50' 
40' 
30' 
2O 

10' 

25° 

10' 

20' 
30' 
40' 
50' 

o  .  4663 

2.1445  65° 

J40° 

0.8391 

1.  1918 

50° 

0.4699 
0.4734 
0.4770 
0.4806 
0.4841 

2.  1283 
2.  1123 
2  .  0965 
2.0809 
2.0655 

so; 

40' 
30' 
20' 

10' 

10' 
20' 
30' 
40' 
50' 

0.8441 
0.8491 
0.8541 
0.8591 
0.8642 

1.1847 
1.1778 

I  .  1708 
I.  1640 
I.  1571 

50' 
40; 
30' 
20' 
10' 

33° 

o  .  6494 

1-5399  J57° 

10' 
20' 

30' 
40' 
50' 

0.6536 
0.6577 
0.6619 
0.6661 
0.6703 

I  -5301 
1.5204 
1.5108 
I  -5013 
I-49I9 

50' 
40' 
30' 

20' 
10' 

26° 

0.4877 

2  .  0503  64° 

41° 

0.8693 

1.1504 

49° 

10' 
20' 
30' 
40' 

SG' 

0.4913 
0.4950 
0.4986 
0.5022 
0.5059 

2.0353 
2  -O2O4 

2.0057 

1.9912 
1.9768 

So'| 

40' 
30' 

20' 
10' 

10' 
20' 
30' 
40' 

50' 

0.8744 
0.8796 
0.8847 
0.8899 
0.8952 

1.1436 
1.1369 
1.1303 
I.  1237 
I.  1171 

so; 
40' 
30' 

20' 
10' 

34° 

0.6745 

1.4826 

56° 

10' 

20' 
30' 
40' 

So' 

0.6787 
0.6830 
0.6873 
0.6916 
0.6959 

1-4733 
1.4641 
1-4550 
I  .  4460 
1-4370 

50' 

40' 
30' 

20' 
10' 

27° 

0.5095 

1.9626  63° 

42° 

0.9004 

i  .  1106 

48° 

10' 

20' 
30' 
40' 
So' 

0.5132 
o  .  5  i  69 
0.5206 
0.5243 
0.5280 

1.9486 

1-9347 
1.9210 
1-9074 
I  .  8940 

So' 

40' 
30' 
20' 
10' 

10' 

20' 

30' 
40' 
50' 

0.9057 
0.9110 
0.9163 
0.9217 
0.9271 

I.  1041 
1.0977 
I.09I3 
I  .0850 
1.0786 

50' 
40' 
30' 

20' 
10' 

35° 

o.  7002 

1.4281 

55° 

10' 
20' 
30' 
40' 
So' 

o.  7046 
0.7089 
0.7133 
0.7177 
0.7221 

I-4I93 
1.4106 
1.4019 
1-3934 
1.3848 

So' 
40' 
30' 
20' 
10' 

28° 

0.5317 

1.8807  62° 

43° 

0.9325 

1.0724  47° 

10' 
20' 
30' 
40' 

So' 

0.5354 
0-5392 
0.5430 
0.5467 
0-5505 

1.8676 
1-8546 
1.8418 
1.8291 
1.8165 

50' 
40' 
30' 

20' 

10' 

10' 

20' 
30' 
40' 

So' 

0.9380 
0.9435 
0.9490 
0.9545 
0.9601 

I.  0661 
i  -  0599 
1.0538 
1.0477 
i  .0416 

SO' 
40' 
30' 

20' 
10' 

36° 

0.7265 

1-3764  54° 

10' 

20' 
30' 
40' 

So' 

0.7310 
0.7355 
0.7400 
0.7445 
0.7490 

1.3680 
1-3597 
I-35I4 
1-3432 
I-  3351 

SO' 

40' 
30' 
20' 

10' 

29° 

0.5543 

I  .  8040 

61° 

44° 

0.9657 

1.0355 

46° 

10' 
20' 
30' 

4C' 
So' 

0.5581 
o.  5619 
o.  5658 
0.5696 
0.5735 

I-79I7 
1.7796 
1.7675 
1.7556 
1-7437 

So' 

40' 
30' 
2O' 

10' 

10' 

20' 
30' 
40; 

So' 

0.9713 
0.9770 
0.9827 
0.9884 
0  .  9942 

1.0295 
1-0235 
I  .0176 
I.OII7 
1.0058 

50' 
40' 

30' 
20' 

10' 

37° 

0.7536J  1.3270  53° 

10' 

20' 
30' 

0.7581 
0.7627 
0.7673 

I-3I90 
I  -3III 
1-3032 

50' 
40' 
30' 

30° 

0-5774 

I-732I 

60° 

45° 

I  .  0000 

I  .  0000 

45° 

Cot    Tan   ;  A 

Cot 

Tan 

A  il 

Cot 

Tan 

A 

INDEX 


Names  in  the  General  Bibliographies  have  not  been  indexed 


0,91 

a,  Crystallographic,  i 

a,  Optic,  91 

Abbe,  E,  131,  132,  133,  180,  187 

Abbe  drawing  apparatus,  297 

Abbe  test  plate,  187 

Aberration,  129 

Abnormal  birefringence,  359 

Absorption  axes,  322 

Absorption  coefficient,  326 

Absorption  of  light,  320 

Accession  catalogue,  609 

Accessories,  Slot  for,  148 

Accessories,  Testing,  231 

Achromatic  lenses,  130 

Active  substances,  108 

Acute  bisectrix,  105 

Adams,  W.  G.,  305 

Adjustment,  Coarse,  149 

Adjustment,  Fine,  149 

Ady,  J.  E.,  574,  585,  593,  594 

Ahrens,  C.  D.,  171 

Ahrens'  prism  (1884),  171 

Ahrens'  prism  (1886),  171 

Airy,  G.  B.,  363 

Alcohol,  Formula  for  mixing,  625 

Allochromatic  colors,  309 

Amann,  J.,  379 

Amann  birefractometer,  379 

Ambronn,  H.,  253,  254,  266 

Ambronn's  method  for  determining 

live  indices,  253,  254 
Amici,  M.,  450 
Amici-Bertrand  lens,  450 
Amorphous  substances,  i,  no 
Amplitude,  33,  35,  49 
Analyzer  and  polarizer,  176 
Anastigmatic,  130 
Andrews,  T.,  573 
Angle,  Extinction,  102 
Angle,  Incidence,  51 
Angle,  Measurement  of,  293 
Angle,  Optic,  102 
Angle,  Polarization,  58 


Angle  of  reflection,  51 

Angle  of  refraction,  59 

Angular  aperture,  131 

Angular  velocity,  33 

Angular  velocity,  Equation  of,  33 

Anisometric  system,  2 

Anisotropic  media,  48,  61,  89,  no 

Anlauffarben,  286 

Anomalies,  Optical,  508 

Anomalous  birefringence,  359 

Anomalous  dispersion,  442 

Anorthic  system,  3 

Anterior  focal  plane,  139 

Antipodal  points,  6 

Anthony,  J.,  152,  299 

Apatite,  Chemical  reactions  on,  565 

Aperture  of  lenses,  13 1 

Aperture  table,  132,  190 

Apertometer,  132 

Aplanatic  lenses,  130 

Apochromatic  lenses,  131 

Apparent  optic  axial  angle,  102 

Arago,  F.  J.,  108,  337,  362,  387 

Aragonite,  Separation  from  calcite,  568 

Arago' s  law,  60 

Areas,  Measurement  of,  290 

Arons,  Leo,  311 

Arschinow,  Wladimir,  307 

Astigmatism  of  lenses,  130 

Asymmetric  system,  3 
refrac-      Attachable  mechanical  stage,  144 

Attractive  minerals,  457 

Augitic  system,  2 

Automatic  section  grinding  machine,  591 

Auxiliary  circle,  32 

Axes  of  ease  of  vibration,  91 

Axes  of  elasticity,  91 

Axes  of  optical  ellipsoid,  61 

Axes  of  optical  ellipsoid,  Locating  position 
in  crystal  by  means  of  a  rotation  ap- 
paratus, 497 

Axes  of  vibration,  61 

Axes  of  vibration,  biaxial,  91 

Axial  angle,  See  also  Optic  axial  angle 

631 


632 


INDEX 


Axial  angle,  102 

Axial  angle  diagram,  471,  491 

Axial  angle,  Equation  for  true,  103 

Axial   angle,    Relation    between    true   and 

apparent,  104,  466 
Axial  angle  scale,  469,  471 
Axial  plane,  Crossed  dispersion  of,  444 
Axis  of  isotropy,  89 
Axis  of  a  lens,  114 
Axis  of  no  double  refraction,  64 
Axis  of  single  wave  velocity,  100 
Axis,  Optic,  64 

fr»  91 

ft,  Crystallographic,  i 
ft,  Optic,  92 
Babinet,  M.,  322,  387 
Babinet  compensator,  373 
Baker's  lamp,  224 
Balance,  Hydrostatic,  515 
Balance,  Jolly,  516 
Balance,  Roger's,  516 
Balance,  Westphal,  533 
Barium  mercuric  iodide  solution,  524 
Bartholinus,  Erasmus,  62 
Bausch    &  Lomb,   Instruments    manufac- 
tured  by,  137,  144,  147,  154,  184,  212, 

227,  298,  613 
Bausch,  Edward,  187 
Beasley,  H.  C.,  589 
Beck,  Instruments  manufactured  by,  217, 

221 
Becke,   F.,   271,    275,    278,  374,  425,  426, 

429,  453,  463,  464,  468,  476,  478,  480, 

483,  484,  568 

Becke-Exner  mikrorefractometer,  275  ' 
Becke-Klein  magnifier,  453 
Becke  line,  277 

Becke  rotating  drawing  stage,  478 
Becke  method  for  determining  axial  angles 

graphically,  476 
Becke  method   for  determining   feldspars, 

278 
Becke   method   for  determining   refractive 

indices,  271 
Becke  method  for  determining  2E  by  means 

of  the  curvatures  of  the  isogyres,  480,  485 
Becke's  graphical  solution  of  sin  E  =  n  sin  V, 

468 

Becker,  G.  F.,  284 
Beckenkamp,  J.,  172,  429,  497 
Beckmann,  E.,  317 


Becquerel,  Edmond,  241 
Becquerel,  H.,  322 

Becquerel  et  Cahors  method  for  determin- 
ing refractive  indices,  241 
Beer,  August,  429 
Behr,  J.,  531 
Behr's    method    for    determining   .specific 

gravity,  531 
Behrens,  H.,  563 
Behrens,  Wm.,  260,  284 
Bell,  Louis,  313 
Bellevue,  F.  de,  572 
Bennett,  A.,  29 
Berek,  Max,  397 
Berkeley,  Earl  of,  517 
Bernhard,  Wilh.,  299 
Bertin,  A.,  242 
Bertin's  method  for  determining  refractive 

indices,  242 
Bertrand,  Emile,  169,  251,  301,  394,  449, 

450,  451,  527 

Bertrand  immersion  fluid,  251 
Bertrand  lens,  178,  450 
Bertrand  lens,  Centering,  230 
Bertrand  method  for  observing  interference 

figures,  449,  451 
Bertrand  ocular,  394 
Bertrand  ocular,  Testing,  230 
Bertrand  prism,  169 
Biaxial  crystals,  91,  93 
Biconcave  lenses,  114 
Biconvex  lenses,  114 
Binormals,  93,  100 
Biot,  J.  B.,  70,  108,  347,  351,  363,  365,  366, 

386,  406,  457,  461,  481 
Biot-Klein  plate,  see  Biot  quartz  plate 
Biot  quartz  plate,  386 
Biot  sensitive  violet,  386 
Biradials,  99 
Birefringence,  61 
Birefringence,  Abnormal,  359 
Birefringence,  Calculation  of,  351 
Birefringence,  Determination  of,  3(^9 
Birefringence,  Determination  of  by  means 

of  rotation  apparatus,  504 
Birefringence,  Lines  of  equal,  355 
Birefringence,  Table  of  maximum,  373 
Bisectrices,  Dispersion  of,  412 
Bisectrices,    Dispersion    of    in    monoclinic 

crystals,  445 
Bisectrix,  105 
Blackburn,  W.,  131 


INDEX 


633 


Blackham,  G.  E.,  180 

Body  tube,  145 

Boeke,  H.  E.,  5 

Booth,  M.  A.,  603 

Borders  in  minerals,  Colored,  249,  258 

Borders  in  minerals,  Dark,  257 

Bornemann,  J.  G.  and  L.  G.,  588 

Bosscha,  J.,  599 

Boyle,  Robert,  572 

Brace,  D.  B.,  388 

Brace's  half-shade  polarizer,  388 

Brandao,  See  Souza-Brandao 

Brauns,  R.,  251,  525,  526 

Brauns'  use  of  methylene  iodide,  251,  525 

Bravais,  A.,  363,  374,  393 

Bravais  twinned  mica  plate,  387,  393 

Bravir,  H.  L.,  285 

Breon,  R.,  545 

Brewster,  Sir  David,  241,  284,  363,  448, 
457,  460 

Brewster  lens,  136 

Brewster's  method  for  determining  refrac- 
tive indices,  241 

Brogger,  W.  C.,  301,  551 

Brogger's  microgoniometer,  301 

Brogger's  separation  apparatus,  551 

Brucite,  Chemical  reactions  on,  567 

Briicke,  E.,  311 

Bryan,  G.  H.,  610 

Biicking,  H.,  403 

Bucking's  apparatus  for  showing  effects  of 
pressure,  509 

Bullock's  filar  micrometer,  289 

Butte  granite,  Mechanical  analysis,  292 

Bxa,  105 

Bx0,  105  t 

f,9i 

Cadmium  borotungstate  solution,  521 

Caffyn,  C.  H.,  589 

Cahours,  Auguste,  241 

Calcite,  Chemical  reactions  on,  565 

Calcite,  Double  refraction  in,  62 

Calcite,  Separation  from  aragonite,  568 

Calderon,  L.,  395,  449 

Calderon  plate,  395 

Calkins,  F.  C.,  284 

Camera  lucida,  296 

Campbell,  William,  286 

Canada  balsam,  559 

Canada  balsam,  Index  of  refraction  of,  283 

Cancrinite,  Chemical  reactions  on,  564 


Cap  diaphragm,  152 

Cap  nicol,  176 

Carbonates,  Chemical  reactions  on,  565 

Card  catalogue,  613 

Cardinal  points  of  lenses,  119 

Cartesian  geometric  formulae,  621 

Catalogue,  Card,  613 

Catalogue  of  specimens,  609 

Cathrein,  A.,  325 

Cedar  oil,  To  thicken,  488 

Cement,  Acid  proof,  624 

Cement  for  glass  to  metal,  625 

Cement  for  leather,  625 

Cement  for  specimens,  624 

Cement,  Water  proof,  624 

Cementing  oven,  595 

Centering  objectives,  148 

Central  Scientific'  Co.,  Instruments  made 
by,  107,  225,  240,  471,  515,  517,  533 

Cesaro,  G.,  379,  401,  415 

Cesaro  wedge,  379 

Chalk,  Preparing  sections  of,  599 

Chamberlain,  C.  J.,  224 

Changing  oculars,  228 

Chaulnes,  Due  de,  238 

Chaulnes'  method  for  determining  index  of 
refraction,  238 

Chauvenet,  Professor,  14 

Chelius,  C.,  525 

Chemical  reactions  on  rock  slices,  559 

Chemical  separations,  558 

Chromatic  aberration,  130 

Chromoscope,  311 

Chrustschoff,  K.  von,  376 

Chrustschoff  twin  compensator,  376 

Church,  A.  H.,  519 

Circle  of  confusion,  130 

Circle  of  reference,  32 

Circles  appear  as  true  circles  in  stereo- 
graphic  projection,  8 

Circular  polarization,  107 

Clay,  R.  S.,  129 

Clay,  Thin  sections  of.  599,  601 

Cleavage,  235 

Cleaving,  Method  of,  236 

Clerici,  Enrico,  270,  529 

Clerici's  heavy  fluid,  529 

Clerici's  method  for  determining  refractive 
indices,  270 

Clinorhombic  system,  2 

Clinorhomboidal  system,  3 

Clutch  for  objectives,  227 


634 


INDEX 


Coal,  Thin  sections  of,  601 

Coarse  adjustment,  149 

Coddington  lens,  136 

Coefficient  of  absorption,  326 

Cohen,  E.,  323,  450,  533,  538,  551,  564 

Collection  bags,  606 

Collections  of  petrographic  material,  605 

Collimation,  Line  of,  210 

Color  determination,  310 

Color  of  light,  49,  no 

Color  of  minerals,  309 

Color  of  thin  plates,  328 

Color  scale  according  to  Kraft,  33  2 

Color  scale  according  to  Newton,  330 

Color  scale  according  to  Quincke,  33 1 

Colored  borders  of  minerals,  Cause  of,  249, 
258 

Compensating  oculars,  194 

Compensation,  366 

Compound  microscope,  138 

Concave  lenses,  114  ' 

Condensing  system,  154 

Confusion  circle,  130 

Conical  refraction,  Exterior,  102 

Conical  refraction,  External,  102 

Conical  refraction,  Interior,  100 

Conical  refraction,  Internal,  100 

Conjugate  foci,  Equation  for,  117 

Conjugate  foci  of  convex  lense.,  116 

Conoscope,  413 

Converging  lenses,  114 

Convergence  of  lens,  Equation  of,  122 

Convergent  light,  Observations  by,  413 

Conversion  tables,  weights  and  measures, 
623 

Convex  lenses,  114 

Cordier,  P.,  572 

Corpuscular  theory  of  light,  29 

Correction  collar,  186 

Cosine  table,  626 

Cotangent  table,  628 

Cover-glass,  Correction  for,  186 

Cover-glass,  Compensation  for  in  objec- 
tives, 186 

Cover-glasses,  Effect  of,  185 

Craig,  Thomas,  9 

Crew,  Henry,  30 

Crisp,  Frank,  139,  188,  296 

Critical  angle,  56,  no 

Cross,  W.,  292 

Crossed  axial  plane  dispersion,  444 

Crossed  nicols,  Examination  between,  336 


Crossed  dispersion,  447 

Cross-hairs,  Adjusting,  229 

Cross-hairs,  Focussing  in  ocular,  197 

Cross-hairs,  Replacing,  197 

Crystal,  i 

Crystal  form,  Determining,  233 

Crystal    system,  Determining   by  rotation 

apparatus,  503 
Crystallographic  axes,  i 
Crystallographic  axes,  Relation  of  to  the 

optical  ellipsoid,  390 
Cubic  system,  i 
Curved  ruler,  14 
Curves  of  equal  velocity,  429 
Cylinder  diaphragms,  152 
Czapski,  S.,  187,  208,  288,  297,  413,  453, 

467 
Czapski  ocular,  453 

Dafert,  E.  W.,  541 

Dale,  T.  N.,  285 

Dallinger,  W.  H.,  183 

Daly,  R.  A.,  402 

Dana,  J.  D.,  i,  2 

Dark  borders,  Cause  of,  257 

Decomposed  rock,  Sections  of,  599 

Definition,  Depth  of,  180 

Definition  of  objectives,  180 

Delesse,  A.,  290 

Deleuil,  166 

De  Lorenzo,  See  Lorenzo 

Demonstration  oculars,  196 

Density  to  refractive  index,  Relation  of,  285 

Depth  ol  definition,  180 

Depth  of  focus,  180 

Derby,  Orville  A.,  541 

Deb  Cloizeaux,  A.,  i,  2,  4*46 

DeSouza-Brandao,  See  Souza-Brandao 

Detmers,  H.  J.,  188 

Diamond  saws,  580 

Diaphragms,  151 

Diatom  test  plates,  191 

Dichroism,  320 

Dichroscope,  325 

Dichroscope  ocular,  326 

Dick,  Allan,  177,  215 

Dick  microscope,  215 

Diedrichs,  K.,  316 

Diller.  J.  S.,  553 

Diller  separating  apparatus,  553 

Dilute  colors,  309 

Dimetric  system,  i 


IXDEX 


635 


Dippel,  L.,  191 

Direct  vernier,  145 

Direction  of  vibration  in  uniaxial  crystals, 

73 
Dispersed  white  light  for  monochromatic 

illumination,  317 
Dispersion,  Anomalous,  442 
Dispersion  croisee,  447 
Dispersion,  Crossed,  447 
Dispersion,  Crossed  axial  plane,  444 
Dispersion,    Effect    of   temperature   upon, 

448 

Dispersion,  Horizontal,  446 
Dispersion,  Inclined,  445 
Dispersion  inclinee,  446 
Dispersion  in  monoclinic  crystals,  445 
Dispersion  in  orthorhombic  crystals,  443 
Dispersion  in  triclinic  crystals,  448 
Dispersion,  Influence  upon  extinction  angles, 

412 

Dispersion,  Normal,  442 
Dispersion  of  bisectrices,  412,  445 
Dispersion  of  light,  54 
Dispersion  of  light  in  crystals,  442 
Dispersion  of  optic  axes,  443 
Dispersion,  Selective,  442 
Dispersion  tournnate,  447 
Dispersion,  Un symmetrical,  448 
Distance  of  most  distinct  vision,  133 
Displacement  in  a  circle,  Equation  of,  32 
Distinct  vision,  Distance  of,  133 
Distinct  cleavage,  236 
Diverging  lenses,  114 
Doelter,  C.,  539 
Dolomieu,  D.,  572 

Dolomite,  Chemical  reactions  on,  565 
Double  concave  lenses,  1 14 
Double  lenses,  1 14 
Double  refracting  goniometer,  293 
Double  refraction,  61 
Double  refraction,  Graphical  representation 

of  variation  in  different  directions  in  a 

crystal,  506 

Double  refraction  in  calcite,  62 
Double  refraction  in  calcite,  Apparatus  for 

demonstrating,  69 
Double     refraction     in     uniaxial     crystals 

(Huy gens'  construction),  82 
Doublets,  136 
Doubly  oblique  system,  3 

Dove,  H.  W.,457 
Dowdy,  S.  E.,  227 


Draw  tube,  145 

Drawing  apparatus,  296 

Drawing-board,  Tilting,  298 

Drei-und-einaxige  system,  i 

Dreibrodt,  O.,  554 

Drescher,  W.  A.  E.,  289 

Diessel,  H.,  564 

Drude,  Paul,  442 

Dry  objectives,  183 

Du  Bois,  H.  E.  J.  G.,  316 

Duboscq,  Jules,  387 

Due  de  Chaulnes,  See  Chaulnes 

Dufrenoy,  i 

Duparc,  L.,  326,  331,  356,  374,  407,  487 

Durand,  W.  F.,  114 

€,  64,  89,  in 

E  (extraordinary  ray),  64,  89 

E  (half  the  optic  axial  angle),  102,  112 

Ease  of  vibration  axes,  91 

Ease  of  vibration  curve,  79 

Ea?e  of  vibration  spheroid,  92 

Ease  of  vibration  surface,  Equation  of,  80 

Ebner,  V.  von,  301 

Edinger,  L.,  196 

Edwards,  W.  B.  D.,  521 

Ehlers,  J.,  322,  326 

Ein  -und-einaxige  system,  2 

Ein-und-eingliedrige  system,  3 

Elasticity  axes,  91 

Elasticity  curve,  79 

Elasticity  ellipsoid,  91 

Electromagnet,  538 

Electromagnetic  theory  of  light,  30 

Elliptical  polarization,  107,  312 

Elongation,  optical  character  of,  361 

Emission  theory  of  light,  29 

Enlargement,  Measurement  of,  287 

Entrance  pupil  of  microscope,  138 

Equal  extinction  curves.  410 

Equal  velocity  curves,  429 

Ether,  31 

Evans,  John  W.,  383,  384,  429,  493 

Evans'  double  quartz  wedge,  384 

Evans'  simple  quartz  wedge,  383 

Ewell,  M.  D.,  185 

Exit  pupil  of  microscope,  138 

Exner,  Sigm,  275 

Exterior  conical  refraction,  102 

External  conical  refraction,  102 

Extinction  angles,  339 

Extinction  angles,  Calculating,  403 


636 


INDEX 


Extinction  angles,   Calculation  in  random 

thin  sections,  399 
Extinction  angles,   Graphical  methods  for 

determining,  406 
Extinction  angles,  Influence  of  dispersion 

upon,  412 
Extinction  angle,  Maximum,  Determination 

by  means  of  a  rotation  apparatus,  504 
Extinction  angle,  Measuring,  392 
Extinction  angle  of  a  face,  392 
Extinction  angle  of  a  mineral,  392 
Extinction  angles  on  no  cleavage  plates  of 

pyroxenes,  402 

Extinction,  Curves  of  equal,  410 
Extinction  diagram,  410 
Extinction,  Inclined,  62,  390,  392 
Extinction,  Measuring,  392 
Extinction,  Parallel,  62,  390 
Extinction  positions,  390 
Extinction,  Symmetrical,  391 
Extraordinary  ray,  64,  89 
Eye  lens,  138 
Eyepiece,  138,  193 
Eye  shade,  226 
Eye  to  be  used  in  work,  225 

Fedorow,  E.  von,  4,  14,  16,  18,  25,  173, 
303,  324,  378,  379,  469,  488,  489,  491, 
494,  495,  496,  497,  498,  499,  BOO,  503, 
504,  598 

Fedorow  comparator,  379 

Fedorow  method  for  determining  low  inter- 
ference colors,  378 

Fedorow  mica  comparator,  366 

Fedorow  prism,  173 

Fedorow  slide  boxes,  489 

Fedorow  stage,  303 

Fedorow  stereographic  net,  16 

Fedorow  three  point  compass,  25 

Feldspar,  Separation  from  quartz,  568 

Ferro,  A.  A.,  403 

Field,  Flatness  of,  181 

Field  lens,  138 

Field  of  view,  140 

Field  of  view,  Measuring,  287 

Filtrations,  Microchemical,  581 

Fine  adjustment,  149 

Finishing  slides,  602 

First  order  red  plate,  365 

Fischer,  H.,  309,  310 

Fitzgerald,  G.  F.,  31,  74 

Flatness  of  field,  181 


Fletcher,  L.,  80,  52,  97,  98,  99,  100 

Fletcher's  indicatrix,  uniaxial,  80,  in 

Fletcher's  indicatrix,  biaxial,  92 

Flink,  G.,  301 

Fluids  for  ray  filters,  314,  315,  316 

Focal  distance  of  lens,  120 

Focal  length,  140 

Focal  length,  Equation  for,  125,  127 

Focal  planes  of  lenses,  119,  120 

Focal  point  of  a  lens,  115,  119 

Focus,  Depth  of,  180 

Focus  of  combined  lenses,  118 

Focussing,  227 

Foot  of  microscope,  142 

Forbes,  David,  574,  587,  599 

Ford,  J.,  603 

Formation  of  image  in  compound  micro- 
scope, 138 

Foucault,  Leon,  165 

Foucault  prism,  165 

Fouque,  F.,  439,  539 

Francotte,  P.,  612 

Frankenheim,  M.  L.,  3 

Freda,  G.,  563 

Fresnel,  Augustin,  30,  74,  92,  337,  343,  406, 
481 

Fresnel's  curve  of  elasticities,  79 

Fresnel' s  ellipsoid,  80 

Friable  material,  Sections  of,  599 

Fripp,  H.  F.,  154 

Fuess,  R.,  143,  377,  579,  593,  611 

Fuess,  Instruments  made  by,  14,  25,  26, 
63,  143,  144,  146,  153,  173,  203,  204, 
206,  207,  220,  224,  236,  240,  275,  289, 
290,  291,  294,  301,  302,  303,  304,  305, 
306,  307,  317,  318,  325  326,  367,  374, 
379,  384,  385,  453,  469,  47i,  479,  49°, 
509,  537,  540,  550,  554,  562,  590,  591, 
595,  611 

Fuess  microscopes,  202,  203,  205,  207,  218 

Fuessner,  K.,  158,  166,  168,  175 

Fuessner  prisms,  168 

7,  Crystallographic,  i 
7,  Optic,  92 
Gadolin,  Axel,  516 
Gauss,  K.  F.,  119 
Gauss'  method,  119 
Gebhardt,  W.,  184 
Gelatinizing  minerals,  562 
Gelblum,  S.,  227 
Gelcich,  E.,  9 


IXDEX 


637 


Giesenhagen,  299 

Gifford,  J.  W.,  316 

Giltay,  E.,  133 

Gladstone,  J.  H.,  285 

Glan,  Paul,  167,  326,  389 

Glan  prism,  167 

Glass  polarizing  prisms,  1 74 

Glazebrook,  R.  T.,  31,  62,  74 

Gnomonic  projection,  5 

Goldschmidt,  V.,  5,  9,  250,  515,  519,  532, 

543,  548 

Goldschmidt  separating  apparatus,  548 
Goldschmidt  specific  gravity  indicators,  543 
Goniometer,  Double  refracting,  293 
Goniometer,  Micro-,  301 
Goniometer  occular,  294 
Good  cleavage,  236 
Govi,  G.,  119 
Grabham,  G.  W.,  215,  274 
Grabham's  explanation  of  the  Becke  line, 

274 
Grabham's  improvements  for  microscope, 

215 

Graeff,  Franz  F.,  563 
Grains,  Thin  sections  of,  602 
Grassmann,  J.  G.,  3 
Grating  micrometer  ocular,  289 
Graton,  L.  C.,  286 

Grayson,  H.  J.,  579,  581,  583,  592,  593,  594 
Grayson's  lap,  592 
Great  circle,  5 
Greek  alphabet,  619 
Griffith,  F.  H.,  152,  227 
Grinding  machines,  588 
Grinding  thin  sections,  585 
Grosse,  W.,  172,  176 
Grosse  prism,  172 
Groth,  P.  von,  2,  173,  331,  413,  449,  578, 

579,  589 

Grundlach,  Ernst,  182 
Gylling,  Hj.,  323 

Gypsum  plate,  See  Unit  retardation  plate 
Gypsum  wedge,  365 
Gypsum    wedge    for    determining    optical 

character  of  uniaxial  minerals,  460 

H  (Half  the  optic  axial  angle)  102,  112 
Haidinger,  W.,  2,  310,  325 
Halbschattenapparate,  388 
Half-shade  plates,  388 
Halle,  Bernhard,  174 
Halle,  Gustav,  325 


Halle  prism,  174 

Halos,  Pleochroic,  323 

Hamilton,  Sir  William,  99,  100,  101 

Hammer  belts,  606 

Hammers,  605 

Hanaman,  C.  E.,  595,  603 

Hand  lenses,  136 

Hand  separation,  557 

Hand  specimens,  607 

Harada,  T.,  568 

Harada  tube,  549 

Harker,  Alfred,  402 

Harmonic  curve,  34 

Harmonic  curve,  Equation  of,  35 

Harmonic  curve,  Equation  of  velocity  in,  36 

Harmonic  curves,  Combinations  of,  41 

Harmonic  motions,  Amplitude  equation,  44 

Harmonic  motions,  Combinations  of,  37 

Harmonic  motion,  Simple,  33 

Harris,  W.  H.,  601 

Hartnack,  165 

Hartnack-Prazmowski  prism,  165 

Hastings,  C.  S.f  62 

Hastings  aplanatic  triplet,  138 

Hauenschild,  A.,  553 

Hauenschild  separating  apparatus,  553 

Hausmann,  i,  2 

Haushofer,  K,  562,  563 

Hauswaldt,  Hans,  245 

Hauynite,  Chemical  reactions  on,  563 

Heavy  fluids,  Errors  resulting  from  their 
use,  556 

Heavy  fluids  for  determining  specific 
gravities,  519  to  530 

Heavy  fluids  for  specific  gravity  separations, 
542 

Heavy  fluids,  Specific  gravities  of,  518 

Heavy  fluids,  Tabulation  of  properties  of, 
528 

Heavy  melts,  545 

Hecht,  B.,  467 

Heeger,  W.,  567 

Hemi-orthotype  system,  2 

Hemi-prismatic  system,  2 

Henniges,  L.,  597 

Henniges'  tweezers  for  holding  cover- 
glasses,  597 

Herschel,  J.  F.  W.,  no 

Hexagonal  system,  i 

Highley,  S.,  300 

Hillebrand,  W.  F.,  556 

Hilton,  H.,  5,  435 


638 


INDEX 


Himmelbauer,  A.,  260 

Hinden,  Fritz.,  566 

Hinden  method  for  separating  calcite  from 

dolomite,  567 

Hirschwald,  J.,  142,  202,  291 
Hirschwald  ocular,  291 
Hirschwald  stage,  143 
Hirst,  G.  D.,  192 
Hitchcock,  R.,  604 
Hlawatsch,  C.,  360,  369,  376 
Hockin,  C.,  131 

Homogeneous  immersion  objective,  183 
Hooke,  30 

Horizontal  dispersion,  446 
Horizontal  small  circles,  6 
Hotchkiss,  W.  O.,  272 
Hotchkiss  explanation   of  the  Becke  line, 

272 

Hovermann,  G.,  324 
Hubbard,  L.  L.,  530 
Hubbard's  determination  of  specific  gravity, 

530 

Hull,  29 

Hutchinson,  A.,  18,  25 
Hutchinson's  protractor,  18 
Hutchinson's  three  point  compass,  25 
Huygens,  C.,  30 
Huygens  eyepiece,  193 
Huyghens,  See  Huygens 
Hydromagnesite,   Microchemical  reactions 

upon,  567 
Hydronephelite,    Microchemical    reactions 

upon,  564 

Hydrostatic  balance,  515 
Hydrostatic  float,  534 
Hydrous  minerals,  Sections  of,  602 

Iddings,  J.  P.,  91,  292 
Idiochromatic  colors,  309 
Illuminating  apparatus,  154 
Illuminating  power,  181 
Immersion  fluids,  259 
Immersion  fluids,  Table  of,  260 
Immersion  objectives,  183 
Immersion  oil,  184 
Immersion  oil  bottle,  184 
Immersion  oil,  To  thicken,  488 
Inactive  substances,  108 
Incandescent   gases   for   producing   mono- 
chromatic light,  316,  317 
Incident  light,  51,  233 
Inclined  dispersion,  445 


Inclined  extinction,  62,  390,  392 

Inclined  illumination,  275 

Index  of  refraction,  54 

Index  of  refraction  and  density,  Relation 
between,  285 

Index  of  refraction  determined  by  the  Becke 
method,  271  to  283 

Index  of  refraction  determined  by  immer- 
sion method,  249  to  270 

Index  of  refraction  of  Canada  balsam,  283 

Index  of  refraction  of  fluids,  Determination 
of,  265 

Indicators,  Refractive  index,  268 

Indicators,  Specific  gravity,  542 

Indicatrix,  Equation  of  biaxial,  94 

Indicatrix,  Optical,  80,  1 1 1 

Indices  (Miller),  3 

Indices  of  refraction,  Principal,  94 

Indistinct  cleavage,  236 

Ink  for  writing  on  glass,  625 

Inostranzeff,  A.  von,  311 

Intensity  equation  of  light,  69 

Intensity  equation  of  light  in  interference 
figures,  418 

Intensity  equation  of  polarized  light  pass- 
ing through  one  mineral  plate,  343 

Intensity  equation  of  polarized  light  pass- 
ing through  two  superposed  mineral 
plates,  346 

Intensity  of  light,  49,  no 

Intensity  of  light  emerging  in  uniaxial 
interference  figures,  418 

Intensity  of  light  passing  through  calcite,  67 

Intensity  of  light,  Variation  in,  59 

Interference,  328 

Interference  between  parallel  nicols,  331, 
333,  38o 

Interference  colors,  348 

Interference  of  polarized  light,  337 

Interference  figure,  415 

Interference  figure,  biaxial,  in  section  cut 
perpendicular  to  the  acute  bisectrix, 
420 

Interference  figure,  biaxial,  in  section  cut 
perpendicular  to  the  obtuse  bisectrix,  423 

Interference  figure,  biaxial,  in  section  cut 
perpendicular  to  an  optic  axis,  424 

Interference  figure,  biaxial,  in  sections  in- 
clined to  the  bisectrix,  423 

Interference  figure,  biaxial,  in  sections 
parallel  to  the  plane  of  the  optic  axes, 
425 


INDEX 


639 


Interference  figures,  isotropic  crystals  (Ano- 
malous), 415 

Interference  figures,  Observation  of,  449 

Interference  figures,  Orientation  of  image 
with  respect  to  object,  456 

Interference  figure,  uniaxial,  Cause  of,  416 

Interference  figure,  uniaxial,  in  sections 
parallel  to  the  optic  axis,  419 

Interference  figure,  uniaxial,  in  oblique 
sections,  418 

Interference  sphaerometer,  240 

Interior  conical  refraction,  100 

Internal  conical  refraction,  101 

Iris  diaphragm,  153 

Isochronism,  33 

Isogyres,  417,  421 

Isogyres,  Deduction  from  skiodromes,  434 

Isogyres,  Equation  of,  435,  440 

Isometric  system,  i 

Isotaques,  429 

Isotropic  crystals,  Interference  figures  of, 

4i5 

Isotropic  media,  48,  no 
Isotropy  axis,  64,  89 

Jaggar,  T.  A.,  308 

James,  F.  L.,  603 

Jamin,  M.  J.,  374 

Jellett,  389 

Johannsen,  Albert,   18,  143,  148,  279,  304, 

323,  33i,  367,  455,499 
Johannsen  auxiliary  lens,  454 
Johannsen  drawing-board  for  stereographic 

projection,  18 

Johannsen  quartz-mica  wedge,  367 
Johnsen,  A.,  544,  549 
Johnsen  and  Miigge's  indicators,  544 
Johnston-Lavis,  H.  J.,  600 
Jolly,  P.,  516 
Jolly  balance,  516 
Joly,  J.,  291,  323,  324,  383,  530- 
Joly's    method    for    determining    specific 

gravity  by  immersion  in  paraffine,  530 
Joly's  method  for  measuring  slight  double 

refraction,  383 
Joule,  J.  P.,  517 
Jullien,  J.,  316 

Kalkowsky,  E.,  490,  518,  524 
Karpinskij,  A.,  518,  524 
Keilhack,  K.,  541 
Kirchhoff,  G.,  413 


Kirschmann,  A.,  316 

Klein,  Carl,  148,  252,  284,  302,  305,  306, 

387,  450,  451,  456,  493,  508 
Klein,  Daniel,  521 
Klein  immersion  fluid,  252 
Klein    method    for   observing   interference 

figures,  450 

Klein  quartz  plate,  387,  394 
Klein'sche  Lupe,  453 
Klein  solution,  521 

Klein  solution,  Method  of  preparation,  521 
Klocke,  F.,  508 

Klonne  und  Muller  slide  marker,  612 
Knight,  C.  W.,  286 
Knopf,  A.,  564 
Knorre,  V.,  289 
Knotenpunkte,  120 
Kobell,  F.  von,  i,  2,  394 
Kobell  stauroscope,  394 
Kolk,  S.  v.  d.,  see  Schroeder   v.  d.Kolk 
Koch,  Alfred,  289 
Konigsberger,  J.,  286,  326,  388 
Kongisberger  ocular,  388 
Kraft,  C.,  331,  332 
Krantz,  F.,  Instruments  made  by,  26,  27, 

543,  544 

Kraus,  E.  H.,  448,  517 
Kreider,  D.  A.,  555 
Kreutz,  St.,  486,  568 
Kronig,  Dr.,  604 
Ktenas,  K.  A.,  323 
Kuznitzky,  M.,  196 

7,  49 

L  (Levy),  380 

Labeling  specimens,  608,  609,  610 

Labels  for  thin  sections,  610 

Labels,  Permanent,  609 

Lacroix,  A.,  370,  377,  403 

Lamps,  Microscopic,  223 

Landolt,  H.,  314,  316,  387,  388,  389 

Lang,  V.  von,  325 

Laps  for  section  cutting,  591 

Larsen,  E.  S.,  263,  285 

Lasaulx,  A.  von,  449 

Lasaulx  method  for  observing  interference 

figures,  449 
Laspeyres,   H.,   322,    413,    450,   451,    521, 

553 

Laspeyres  method   for  observing  interfer- 
ence figures,  45 1 

Laspeyres  separating  apparatus,  553 


640 


INDEX 


Lateral  magnification,  Equation  for,  121, 
122 

Lateral  spherical  aberration,  130 

Laurent,  L.,  389 

Least  count,  145 

Lebedew,  29 

Leeson,  H.  B.,  293,  300 

Leeson  prism,  293 

Lehmann,  J.,  575,  589 

Leick,  W.,  517 

Leiss,  C.,  62,  143,  146,  173,  J78,  203,  205, 
207,  218,  240,  275,  289,  301,  302,  303,  305, 
306,319,  325,326,374,376,377,379,453, 
454,  469,  489,  537,  58o,  590,  611 

Leiss  prism,  172 

Leitz,  E.,  149 

Leitz,  Instruments  made  by,  137,  141,  144, 
149,  150,  152,  155,  185,  196,  200,  201, 
224,  296,  297,  414 

Lemberg,  J.,  564,  565,  566,  568 

Lengths,  Measurement  of,  288 

Lenk,  H.,  454 

Lenk-Lasaulx  method  for  observing  inter- 
ference figures,  453 

Lenses,  114 

Lenses,  Care  of,  228 

Lenses,  Formation  of  images  by,  Equation  • 

for,  122 

Lens  stands,  136,  137 
Leo,  Max,  286 
Lepinay,  J.  Mace  de,  374 
Lepinay  half-shade  plate,  396 
Leroux,  E.  P.,  442 
Levy,  380 

Levy,  Michel-,  See  Michel-Levy 
Liebisch,  T.,  101 
Light,  Amount  of,  227 
Light  for  microscopic  work,  223 
Light,  Nature  of,  29 
Limb  of  microscope,  142 
Lincio,  G.,  149,  155 
Linck,  J.,  518,  544,  557,  566 
Lincoln,  F.  C.,  286 
Linebarger,  C.  E.,  517 
Line  of  collimation,  210 
Line  of  equal  birefringence,  355 
Line  of  single  normal  velocity,  100 
Line  of  single  ray  velocity,  99 
Lippich,  F.,  389 
Liquids,  Specific  gravity  of,  518 
Listing,  J.  B.,  119 
Lloyd,  Rev.  H.,  101 


Locating    points  in    stereographic    projec- 
tion, 6 

Lommel,  E.  von,  108,  173,  389,  442 
Lommel  prism,  173 

Longitudinal  spherical  aberration,  130 
Long  tube  microscopes,  140 
Lord,  C.  L.,  601 
Lorentz,  H.  A.,  31,  285 
Lorenz,  L.,  285,  442 
Lorenzo,  G.  de.  262,  277 
Luedecke  separating  apparatus,  554 
Lummer,  O.,  389 

Madan,  H.  G.,  171 

Madan  prism,  171 

Magensite,  Chemical  reactions  on,  565 

Magnification,  Compound  microscope,  197 

Magnification,  Lateral,  121 

Magnifying  power,  133,  140,  182 

Magnifying  power,  Proper  to  use,  227 

Makers  of  microscopes,  199 

Malassez,  L.,  182 

Mallard,  E.,  5,  467 

Mallard's  constant,  468 

Mallard's  formula,  468 

Mallard's  formula,  Accuracy  of,  468 

Mallard's  method  for  measuring  the  optic 

axial  angle,  467 
Malus'  law,  59 
Mann,  Paul,  523,  541,  602 
Mann's  separating  instrument,  541 
Manufacturers  of  microscopes,  199 
Manufacturers  of  thin  sections,  588 
Marking  thin  sections,  6 1 1 
Marpmann,  G.,  255,  263,  594 
Marpmann  immersion  fluid,  254 
Maschke,  O.,  249 

Maschke's  method  for  determining  refrac- 
tive indices,  249 
Maxwell,  J.  Clerk,  31,  74 
Measurement  of  areas,  290 
Measurement  of  enlargement,  287 
Measurement  of  extinction,  392 
Measurement  of  field  of  view,  287 
Measurement  of  lengths,  288 
Measurement  of  plane  angles,  293 
Measurement  of  thicknesses,  293 
Measurements  under  the  microscope,  287 
Measures,  Table  of,  623 
Mechanical  analysis  Butte  "granite,"  292 
Mechanical  separation  of  rock  constituents, 
537 


INDEX 


641 


Mechanical  tube  length,  138 

Mechanical  stages,  142 

Meigen,  W.,  568 

Meigen's  method  for  separating  calcite  from 

aragonite,  568 
Melatope,  420 

Melatopes,  Locating  biaxial,  426 
Melatopes,  Locating  uniaxial,  425 
Melilite,  Chemical  reactions  on,  563 
Meridians,  6 
Merrill,  Geo.  P.,  613 
Merwin,  H.  E.,  263,  264,  534 
Merwin  and  Larsen's  immersion  fluids,  263 
Merwin's  method  for  determining  specific 

gravity  by  refractive  indices  of  fluids,  534 
Methylene  iodide,  525 
Methylene  iodide,  Table  showing  relation 

between  temperature  and  specific  gravity, 

526 

Metz,  C..  149 

Mica  plate,  See  Quarter  undulation  plate. 
Mica  wedge,  366 
Michel-L6vy,  A.,   16,   268,  323,  331,  355, 

370,  377,  403,  406,  411,  439,  472 
Michel-Levy  birefringence  chart,  370 
Michel-Levy  comparator,  377 
Michel-L6vy  method  for  measuring  2E,  472 
Michel-L6vy    refractive    index    indicators, 

268 

Microchemical  nitrations,  561 
Microchemical  reactions,  559 
Microchemical  reactions,  Apparatus  for,  559 
Microchemical    reactions,     Preparing    the 

slide,  560 

Micrometer,  Caliper,  240 
Micrometer  ocular,  288 
Micrometer  ocular,  Grating,  289 
Micrometer  ocular,  Scale,  289 
Micrometer  ocular,  Screw,  289 
Micron,  288 
Microscope,  199 
Microscope,  Compound,  138 
Microscope  lamps,  223 
Microscope  manufacturers,  199 
Microscope,  Mechanical  parts  of,  142 
Microscope,  Optical  parts  of,  154 
Microscope,  Petrographic,  141 
Microscope,  Selecting,  222 
Microscope,  Simple,  136 
Miers,  Henry,  5,  416 
Miller,  W.  H.,  i,  2,  3,  4,  5 
Miller  indices,  3 


Mirror,  157 

Mohl,  Hugo  von.  289 

Mohs,  i,  2 

Moigno,  362 

M oiler,  H.  J.,  310 

M  oiler,  J.  D.,  191 

Molten  substances  for  determining  specific 

gravity,  535 
Monoclinic  system,  2 
Monoclinohedral  system,  2 
Monodimetric  system,  i 
Monosymmetric  system,  2 
Monotrimetric  system,  i 
Monochromatic  light,  313 
Monochromator,  318 
Montigny,  M.  C.,  134 
Mounting  thin  sections,  593 
Miigge,  O.,  323,  324,  544,  549 
Murdoch,  Jos.,  286 
Muthmann,  W.,  529 
Muthmann  heavy  fluid,  529 

Nachet,  A.,  213,  302 

Nachet  camera  lucida,  296 

Nachet,  Instruments  made  by,   154,   213, 

297,  377 

Nagel,  W.A.,  315 
Nageli,  Carl,  300 
Xakamura,  S.,  389 
Xaumann,  C.  F  ,  i,  2,  3 
Naumann  parameters,  3 
Negative  biaxial  crystals,  105 
Negative     character,     Determination     by 

rotation  apparatus,  503 
Negative  elongation,  33,  362 
Negative  minerals,  Determination  of  optical 

character  of,  457 

Negative  uniaxial  crystals,  70,  in 
Nelson,  E.  M.,  132,  133,  138,  149,  157,  180, 

181,  182,  188 

Nephelite,  Chemical  reactions  on,  564 
Neumann,  F.  E.,  5,  351 
Neutral  curves,  Equation  of,  440 
Newton,  Sir  Isaac,  29,  328,  330 
Newton's  rings,  328 
Newton's  scale  of  colors,  328,  330 
Nichols,  29 
Nicol,  W.,  158 
Nicol,  Cap,  176 
Nicol  net,  434 
Nicol  prism,  158 
Nicol  prism,  Adjusting,  23 1 


642 


INDEX 


Nicol  prism,  Care  of,  228 

Nicol  prism,   Determination  of  vibration 

direction  in,  178 
Nikitin,  W.,  25,  385,  504 
Nikitin  hemisphere,  26 
Nikitin's    method    for    measuring    slight 

double  refraction,  383 
Nikitin's  quartz  compensator,  385 
Nodal  points  of  lenses,  119,  120 
Noll,  F.,  19 

Normal  dispersion,  442 
Northrup,  Zae,  610 
Noselite,  Chemical  reactions  on,  563 
Nose  piece,  147 
Nowacki,  A.,  541 
Numerical  aperture,  131 
Numerical  aperture  table,  132 
Nutting,  P.  G.,  311 

O,  64,  89 

co,  64,  89,  ill 

Object  clips,  142 

Objective,  138,  180 

Objective  clutch,  227 

Objective  holder,  146 

Objectives,  Classification  of,  185 

Objectives,  Comparative  table  of,  189 

Objectives,  Cost  of,  193 

Objectives,  Testing,  191 

Objectives  with  correction  collar,  186 

Object  lens,  138 

Oblique  system,  2 

Obtuse  bisectrix,  105 

Ocular,  138,  193 

Ocular,  Bertrand,  394 

Ocular,  Bertrand,  Testing,  230 

Ocular,  Czapski,  453 

Ocular,  Demonstration,  196 

Ocular  dichroscope,  3  26 

Ocular  goniometer,  294 

Ocular,  Schwarzmann's,  470 

Oculars,  Comparative  table  of,  195 

Oculars  for  special  purposes,  195 

Oebbeke,  K,  549,  550 

Oebbeke  tube  for  specific  gravity  deter- 
minations, 550 

Office  work,  609 

Olivine,  Chemical  reactions  on,  565 

One-fourth  order  mica  plate,  See  Quarter 
undulation  plate. 

Opaque  minerals,  Color  determined  by 
means  of  a  rotation  apparatus,  503 


Opaque  minerals,  Determination  of  color 
of,  311 

Opaque  minerals,  Examination  of,  285 

Opening  angle  of  nicol,  162 

Optical  anomalies,  508 

Optical  character  of  elongation,  361 

Optical  character  of  a  mineral,  457 

Optical  character  of  a  mineral,  Determina- 
tion by  rotation  apparatus,  503 

Optical  center  of  lens,  114 

Optical  curves,  494 

Optical  ellipsoid,  80,  1 1 1 

Optical  ellipsoid,  Axes  of,  61 

Optical  ellipsoid,  Relation  of,  to  crystallo- 
graphic  axes,  390 

Optical  indicatrix,  80,  in 

Optical  parts  ot  a  microscope,  154 

Optical  tube  length,  138 

Optic  axes,  Dispersion  of,  443 

Optic  axes,  Plane  of,  105 

Optic  axis,  64 

Optic  axis,  Biaxial,  93,  99,  100 

Optic  axis,  Biaxial,  Locating  the  point  of 
emergence,  426 

Optic  axis,  Determination  by  rotation 
apparatus  in  sections  nearly  parallel  to 
the  plane  of  the  optic  axes,  499 

Optic  axis,  Determination  of  point  of 
emergence,  476 

Optic  axis,  Locating  point  of  emergence  in 
uniaxial  crystals,  425 

Optic  axis,  Locating  one  by  optical  curves, 

494 
Optic  axis.  Locating  one  by  rotation  stage, 

489 
Optic  axis,  Locating   second   by   rotation 

stage,  495 
Optic  axis,  Locating  second  when  first  has 

been     determined     by    optical    curves, 

498 

Optic  axis,  Primary,  93,  100 
Optic  axis,  Secondary,  99 
Optic  axis,  Simplified  method  for  locating, 

500 

Optic  axis,  Uniaxial,  70 
Optic  angle,  See  Optic  axial  angle. 
Optic  axial  angle,  102 
Optic  axial  angle,  Apparent,  102 
Optic  axial  angle,  Equation  for  true,  103 
Optic  axial  angle,  Measurement  by  means 

of  a  rotation  stage,  487 
Optic  axial  angle,  Measurement  of,  466 


INDEX 


643 


Optic  axial  angle,  Relation    between    true 

and  apparent,  104 
Optic  axial  angle,  True,  102 
Optic  binormals,  93,  100 
Optic  biradials,  99 
Optic  section,  Principal,  70,  92 
Ordinary  light,  233 
Ordinary  ray,  64,  89 
Orientation,  361 
Orthographic  projection,  5 
Orthorhombic  system,  2 
Orthoscopic  lenses,  130 
Orthotype  system,  2 
Osann,  A.,  564 
Oschatz,  Dr.,  573 
Oscillation  in  ether,  3 1 
Over-corrected  lenses,  130 

Packing  specimens  for  shipment,  608 

Panebianco,  G.,  568 

Parallel  extinction,  62,  390 

Parallels,  6 

Parameters,  Weiss,  3 

Parker,  C.  B.f  604 

Parkes'  microscope  lamp,  223 

Parting,  235 

Passage  of  light  through  two  nicols  and  a 
mineral  plate,  341 

Pauly,  Anton,  266 

Pauly's  method  for  determining  the  re- 
fractive indices  of  fluids,  266 

Pearce,  F.,  326,  356,  374,  406,  407 

Pearcey,  F.  G.,  601 

Pebal,  L.,  540 

Peiser,  J.,  226 

Penetration  of  objectives,  180 

Penfield,  S.  L.,  u,  16,  17,  22,  516,  546,  555 

Penfield's  protractor,  16 

Penfield's  separating  apparatus  for  heavy 
melts,  555 

Pennock,  E.,  226 

Perfect  cleavage,  235 

Period,  33 

Periodic  motion,  32 

Petri,  R.  J.,  138 

Petrographic  microscope,  141 

Pfaff,  F.,  599 

Pfitzner,  W.,  603 

Phase,  33 

Phasal' difference,  Equation  of,  337 

Pillar  of  microscope,  142 

Pirsson,  L.  V.,  292 


Plane  of  optic  axes,  105 

Plane  of  polarization,  58 

Plane  polarized  light,  58,  107 

Planimeter  ocular,  291 

Plano-concave  lenses,  1 14 

Plano-convex  lenses,  114 

Playfair,  Lyon,  517 

Pleochroic  halos,  323 

Pleochroism,  320 

Pleochroism,  Determination  of,  325 

Polarimetre  a  pSnombre,  388 

Polariscope,  413 

Polarizer  and  analyzer,  176 

Polarization,  57 

Polarization  angle,  58 

Polarization  by  double  refraction,  106 

Polarization  by  reflection,  57 

Polarization  by  refraction,  59 

Polarization,  Circular,  107 

Polarization,  Elliptical,  107 

Polarization  of  light  by  lenses,  415 

Polarization  plane,  58,  107 

Polarizing  prisms,  159 

Polarizing  prisms,  Properties  of,  175 

Pole,  6 

Polishing  rocks,  602 

Poor  cleavage,  236 

Porous  substances,  Specific  gravity  deter- 
mination of,  518 

Position  for  work,  225 

Positions  of  extinction,  .390 

Positive  biaxial  crystals,  105 

Positive  character  determined  by  rotation 
apparatus,  503 

Positive  elongation,  33,  362 

Positive  minerals,  Determination  of  char- 
acter, 457 

Positive  uniaxial  crystals,  70,  in 

Posterior  focal  plane,  138 

Post  of  microscope,  142 

Potassium  mercuric  iodide  solution,  Use  of, 

Si9 

Powders,  Thin  sections  of,  60 1 
Prazmowski  prism,  165 
Preston,  T.,  374,  442 
Pribram,  316 

Primary  optic  axes,  93,  100 
Principal  focal  point,  115 
Principal  focal  point,  First,  120 
Principal  focal  point,  Second,  120 
Principal  focus,  combined  lenses,  Equation 

for,  119 


644 


INDEX 


Principal  focus,  Equation  for,  118 
Principal  indices  of  refraction,  94 
Principal  optic  section,  biaxial,  91 
Principal  optic  section,  uniaxial,  70 
Principal  points  of  lenses,  119,  120 
Principal  vibration  axes,  61 
•  Pringsheim,  E., -3 16 
Prismatic  system,  2 
Properties  of  polarizing  prisms,  175 
Protractors  for  stereographic  projection,  14, 

16,  18 

Protractor,  Hutchinson,  18 
Protractor,  Penfield,  16 
Pseudo-absorption,  324 
Pseudo-dichroism,  324 
Pseudo-pleochroism,  324 
Pumice,  Thin  sections  of,  600 
Pupil  of  eyepiece,  138 
Pycnometer,  517, 
Pyramidal  system,  i 
Pyroxenes,  Extinction  angles  in,  402 

Quadratic  system,  i 

Quarter  order  mica  plate,  see  Quarter  un- 
dulation plate 

Quarter  undulation  plate,  362 

Quarter  undulation  plate  for  determining 
optical  character  of  biaxial  minerals,  462 

Quarter  undulation  plate  for  determining 
optical  character  of  uniaxial  minerals,  457 

Quartz,  Separation  from  feldspar,  micro- 
chemically,  568 

Quartz  wedge,  365 

Quartz  wedge  for  determining  optical  char- 
acter of  biaxial  minerals,  462 

Quartz  wedge  for.  determining  optical 
character  of  uniaxial  minerals,  460,  461, 
462 

Queckett,  196 

Quincke,  G.,  331,  374 

Radde's  color  scale,  3 1^ 

Radioactivity,  323 

Ramsay,  W.f  322 

Ramsden  disk,  138 

Ramsden  eyepiece,  194 

Rapp,  521 

Rath,  G.  vom,  564 

Rauff,  H.,  579,  592 

Ray,  Extraordinary,  64,  89 

Ray  filters,  314. 

Ray  front,  50 


Rayleigh,  Lord,  3 1 

Ray,  Ordinary,  64,  89 

Ray  surface,  Biaxial,  94 

Ray  surface,  Biaxial,  Equation  of,  97 

Ray  surface,  Uniaxial,  Equation  of,  75 

Ray  surface,  Uniaxial,  Graphical  develop- 
ment of,  76 

Reactions,  Microchemical,  559 

Real  focus,  115 

Recipes,  Useful,  624 

Red  of  first  order,  See  Unit  retardation 
plate 

Reflection  of  waves,  50 

Refraction,  52 

Refraction,  Double,  61 

Refraction,  Double  in  calcite,  62 

Refraction,  Single,  61 

Refraction  through  lenses,  116 

Refractive  index,  54 

Refractive  index  and  density,  Relation  be- 
tween, 285 

Refractive  index,  Determining,  237 

Refractive  index,  Determining  by  the  Becke 
method,  271  to  283 

Refractive  index,  Determination  by  rota- 
tion apparatus,  504 

Refractive  index,  Method  for  determining 
by  immersion,  249  to  270 

Refractive  index  of  Canada  balsam,  283 

Refractive  index  of  fluids,  Determination 
of,  265 

Refractive  index,  Principal,  94 

Refractive  index,  Relation  to  velocity  of 
light,  56 

Regnault,  387 

Regular  system,  i 

Reichert,  Instruments  made  by,  151,  196, 
210,  211,  275,  294,  611 

Relief,  237  ' 

Replacing  cross-hairs,  197 

Repulsive  minerals,  457 

Resolving  power,  181 

Retardation,  Equation  for,  337 

Retardation  wedges,  365 

Retgers,  J.  W.,  252,  518,  526,  527,  531, 
546,  547,  602 

Retgers'  determinations  of  specific  gravity, 
526,  531 

Retgers'  heavy  fluids,  526 

Retgers'  immersion  fluids,-252 

Retrograde  vernier,  145 

Rhombic  system,  2 


INDEX 


645 


Rhomohedral  system,  i 

Richthofen,  F.  von,  605 

Ridgeway,  R.,  310 

Rims  around  thin  sections,  602 

Rinne,  F.,  460,  463 

Riva,  C.,  262,  277 

Rogers,  Austin,  516 

Rogers'  specific  gravity  balance,  516 

Rohrbach,  Carl,  251,  524 

Rohrbach's  solution,  524 

Rohrbach's  solution,  Index  of,  251 

Rohrbach's  solution,  Method  of  preparation, 

524 

Rollet,  A.,  331 
Rose,  i 
Rosenbusch,  H.,  99,  237,  284,  323,  331,  374, 

394,  401,  410,  460,  468,  521,  539,  557, 

585,  589 

Rosiwal,  August,  291 

Rosiwal  method  for  measuring  areas,  292 

Rosenhain  microscope,  221 

Rotary  polarization,  108 

Rotary  polarization  in  quartz,  Table  of, 
109 

Rotating  drawing-board,  19 

Rotating  drawing  stage,  478 

Rotating  plane  of  projection  in  stereo- 
graphic  projection,  24 

Rotation  apparatus,  300  to  308 

Rotation  apparatus,  Adjusting,  488 

Rotation  apparatus  for  measuring  2E,  487 

Rowland,  H.  A.,  313 

Royston-Pigott,  182 

Rumpf,  J.,  575 

Sabot,  R.,  487 

Salomon,  W.,  277,  279,  534 

Salomon's  method  for  computing  co  —  e,  383 

Salomon's  method  for  determining  refractive 

indices,  279 
Salomon's  method  for  determining  specific 

gravity,  534 

Sand,  Thin  sections  of,  602 
Sang,  E.,  158,  164 
Sang  prism,  164 
Sarasin,  Ed.,  108 
Sauer,  G.  A.,  563,  564 
Sauter,  F.,  9 
Savart  plate,  386 
Savart's  bands,  386 
Sawing  a  rock  slice,  583 
Scale  micrometer  ocular,  288 


Scales  for  stereographic  projection,  14 

Schaffgotsch,  F.  G.,  519 

Schaller,  W.  T.,  284 

Schieck's  microscope  lamp,  223 

Schiefferdecker,  P.,  226,  611 

Schiemenz,  P.,  296 

Schistoskop,  311 

Schmidt,  K.  E.  F.,  374 

Schneider,  J.,  309 

Schneiderhohn,  H.,  258,  506 

Schonrock,  P.,  396,  398 

Schraf,  A.,  395 

Schroder,  Hugo,  172 

Schroeder  van  der  Kolk,  252,  255,  256,  257, 

260,  267,  306,  324,  451,  529 
Schroeder  van  der  Kolk's  heavy  fluids,  529 
Schroeder  van  der  Kolk's  immersion  fluids, 

255,  256 
Schroeder    van    der    Kolk's    method    for 

observing  interference  figures,  452 
Schroeder  van  der  Kolk's  use  of  inclined 

illumination,  252,  257 
Schiuz,  H.,  175 
Schuster,  Arthur,  442 
Schuster,  Max,  392 
Schwarzmann,  Max,  469,  470 
Schwarzmann's  axial  angle  scale,  469 
Schwarzmann's  ocular,  470 
Schwendener,  S.,  300 
Screw  micrometer  ocular,  289,  290 
Sechsgliedrige  system,  i 
Secondary  optic  axes,  99 
Section  boxes,  612 
Section  cutting  machines,  574 
Section  grinding  machines,  588 
Section  markers,  611 

Seibert,  Instruments  made  by,  202,  288,  537 
Seidentopf,  H.,  384 

Seidentopf  quartz  wedge  compensator,  384 
Seiffert,  Dr.,  597 
Seiler,  C.,  595 
Selecting  a  microscope,  222 
Selective  dispersion,  442 
Sellers,  C.,  588 
Senarmont,  H.  de,  363,  366 
Sensitive  red,  349 
Sensitive  tint,  349 
Sensitive  violet,  349 
Separating  apparatus,  547 
Separating  apparatus  for  heavy  melts,  554 
Separating  by  chemical  means,  558 
Separating  by  hand,  557 


646 


INDEX 


Separating  funnel,  550 

Separating  thin  flakes  and  needles,  557 

Shadbolt,  131 

Shade  for  eyes,  226 

Shagreen  surface,  237 

Short  tube  microscope,  140 

Sigsbee,  C.  D.,  14 

Simple  microscope,  136 

Simultaneous  rotating  nicols,  177 

Sine  table,  626 

Single  refraction,  61 

Skiodromes,  430 

Skiodromes,  To  construct  for  random  sec- 
tions,  433 

Slavik,  F.,  285 

Sleeman,  P.,  169 

Sliding  diaphragm,  152 

Small  circles,  6 

Smeeth,  W.  F.,  517,  552 

Smeeth's  method  for  determining  specific 
gravity,  517 

Smeeth's  separating  apparatus,  552 

Smith,  Hamilton,  604 

Smith,  H.  L.,  262 

Smith,  Herbert  G.  F.,  5 

Smith's  method  for  determining  the  re- 
fractive indices  of  fluids,  265 

Snell's  law,  54 

Societ^  Genevoise,  microscope  made  by,  218 

Sodalite,  Chemical  reactions  on,  563 

Sokol,  R.,  568 

Soleil,  Henri,  387 

Soliel  bi-quartz  plate,  387 

Sollas,  W.  J.,  291,  533,  534 

Sollas  hydrostatic  float,  534 

Sollas  modification  of  the  Sprengel  tube,  533 

Soluble  minerals,  Thin  sections  of,  602 

Soluble  substances,  Determination  of  spe- 
cific gravity  of,  518 

Sonstadt,  E.,  519 

Sonstadt  solution,  519  I 

Sonstadt  solution,  MetHod  of  preparing,  519 

Sommerfeldt,  E.,  153,  177,  397,  429,  454 

Sommerfeldt  condenser,  454 

Sommerfeldt  twinned  gypsum  plate,  388, 

397 
Sorby,  H.  C.,  244,  247,  250,  366,  572,  573, 

596,  598,  599 
Sorby's  method  for  determining  refractive 

indices,  244 

Sorby's  method  for  showing  relief,  250 
Soret,  J.  L.,  108 

-V 


Souza-Brandao,  V.  de,  177,  205,  268,  404, 

Souza-Brandao  axial  angle  diagram,  471 
Souza-Brandao  refractive  index  indicators, 

268 

Spassky,  M.,  158 

Specific  gravity  determination,  515 
Specific  gravity  determinations  by  means 

of  heavy  fluids,  532 
Specific  gravity  determinations  by  means  of 

molten  substances,  535 
Specific  gravity  indicators,  542 
Specific  gravity  separations  by  means  of 

heavy  fluids,  542 
Specific  gravity  separations  by  means  of 

water,  541 

Specific  gravity  table,  544 
Specimens,  607 
Spezia,  G.,  309 
Sphaerometer,  240 
Spherical  angles  appear  in  their  true  values 

in  stereographic  projection,  9 
Sprengle  tube,  532 
Sprockhoff,  M.,  285 
Stages,  Microscope,  142 
Staining  minerals,  562 
Stand,  Care  of,  228 
Stanhope  lens,  136 
Stark,  Michael,  285,  485 
Stark's  modification  of  Becke's  method  for 

determining  2E.,  485 
Stauroscope,  394 
Steeg  und  Reuter,  Instruments  made  by, 

59,  3i7,  590,  612 
Steenstrup,  K.  J.  V.,  589,  599 
Steinheil  lens,  136 
Steinmann,  G.,  576,  581,  583,  593 
Steinmann's  section  cutting  machine,  576 
Steinriede,  541 
Stelzner,  A.,  564,  565 
Stephenson,  J.  W.,  181,  191,  251 
Stereographic  projection,  5 
Stereographic  projection,  Accuracy  of,   22 
Stereographic  projection  net,  16,  17 
Stevens,  W.  LeC.,  287 
Stigmatic  lenses,  130 
Stober,  F.,  396 

Stober  quartz  double  plate,  396 
Stoe,  Instruments  made  by,  543 
Stokes,  G.  G.,  246 
Stolze,  174 
Story-Maskelyne,  5 


INDEX 


647 


Streng,  A.,  530,  561,  564,  565 

Streng's  determination  of  specificgravity,53O 

Strutt,  R.  J.,  324 

Subnormal  color,  360 

Substage,  150 

Supernormal  colors,  360 

Surface  d'elasticite,  92 

Swift,  Instruments  made  by,  214,  216 

Symmetrical  extinction,  391 

Symmetry  planes,  Locating,  497 

Table,  225 

Tait,  P.  G.,  158,  104,  442 
Talbot,  H.  F.,  167,  442 
Talbot  prism,  167 
Tangent  table,  628 

Teall,  J.J.H.,541 

Teinte  de  passage,  365 

Teinte  sensible,  365 

Temperature,  Effect  upon  dispersion,  448 

Ten  Siethoff,  E.  G.  A.,  307,  422 

Tertsch,  H.,  485,  486 

Tertsch's  modification  of  Becke's  method 
for  determining  2E,  485 

Tesseral  system,  i 

Tessular  system,  i 

Testing  cross-hairs,  229 

Test  plate,  Abbe,  187 

Test  plate,  Diatom,  191 

Tetarto  prismatic  system,  3 

Tetragonal  system,  i 

Thick  edge  lenses,  114 

Thickness,  Measuring,  293 

Thickness  of  a  lens,  1 14 

Thickness  of  a  lens,  Equation  for,  126 

Thin  edge  lenses,  1 14 

Thin  flakes,  Separating,  557 

Thin  section  boxes,  612 

Thin  sections,  Preparing,  572 

Thompson,  S.  P.,  167,  172 

Thompson  prisms,  167 

Thompson,  J.  J.,  31 

Thoulet,  J.,  237,  250,  519,  535,  547,  602 

Thoulet's  determination  of  the  specific 
gravity  of  minerals  heavier  than  the  im- 
mersion fluid,  535 

Thoulet's  method  for  determining  refrac- 
tive indices,  250 

Thoulet's  separating  apparatus,  547 

Thoulet's  solution,  519 

Thoulet's  solution,  Method  of  preparation, 


Thoulet's  solution,  Relation  between  speci- 
fic gravity  and  refractive  index,  250 

Three  point  compass,  25 

Thiirach,  Hans,  541 

Tilting  drawing-board,  298 

Tinne,  F.,  415 

Tomlinson,  W.  H.,  maker  of  sections,  588 

Tornebohm,  A.  E.,  564 

Total  reflection,  56 

Toula,  Franz,  516 

Tourmaline,  Absorption  by,  106 

Tourmaline  tongs,  107 

Transmitted  light,  233 

Traube,  H.,  323,  396 

Traube  bi-mica  plate,  396 

Triclinic  system,  3 

Triclinohedral  system,  3 

Trigonal  system,  2 

Trigonometric  formulae,  619 

Trimetric  system,  2 

Triplets,  136 

True  optic  axial  angle,  102 

Tschermak,  G.,  310,  325,  413,  516 

Tube  length,  138 

Tube  length  of  various  microscopes,  140 

Tube  of  microscope,  145 

Tutton,  A.  E.  H.,  318,  326,  374,  444,  448, 
593 

Under  corrected  lenses,  130 

Undulatory  theory  of  light,  30 

Unger,  Professor,  573 

Uniaxial  crystals,  62,  70 

Unit  retardation  plate,  349,  365,  393 

Unit  retardation  plate  for  determining  opti- 
cal character  of  uniaxial  crystals,  459, 461, 
462 

Unit  retardation  plate  for  determining  opti- 
cal character  of  biaxial  crystals,  462 

Unsymmetrical  dispersion,  448 

V  (Half  the  optic  axial  angle),  102,  112 

Valentin,  G.,  300,  362 

Van  Heurck,  H.,  191,  312 

Van  Werveke,  L.,  520,  550 

Van  Werveke  separating  funnel,  550 

Velocity  around  a  circle,  Equation  of,  32 

Velocity  of  light,  49 

Velocity  of  ray  in  biaxial  crystals,  Equation 
of,  98 

Velocity  of  ray  in  uniaxial  crystals,  Equa- 
tion of,  71 


648 


INDEX 


Velocity  of  wave  in  uniaxial  crystals, 
Equation  of,  72 

Velocity,  Relation  to  refractive  index,  56 

Verniers,  144 

Vertex  of  a  lens,  1 14 

Vertical  great  circle,  6 

Vertical  small  circle,  6 

Vesicular  rocks,  Thin  sections  of,  600 

Vibration  axes,  61 

Vibration  axes,  Biaxial,  91 

Vibration  directions  in  minerals,  Deter- 
mination of,  361 

Vibration  directions  in  nicol  prisms,  Deter- 
mination of,  178 

Vibration  directions,  Uniaxial,  73 

Vibration  ease,  Curve  of,  79 

Viergliedrige  system,  i 

Viola,  C.,  274,  276,  426,  474 

Viola-Becke  method  for  determining  re- 
fractive indices,  276 

Viola-Becke-de  Chaulnes  method  for  deter- 
mining refractive  indices,  276 

Viola  method  for  determining  2E,  474 

Violet  of  first  order,  See  Unit  retardation 
plate 

Virtual  image,  115 

Vogelsang,  H,  564,  574 

Voigt,  W.,  322 

Voigt  und  Hochgesang,  Makers  of  sections, 
588 

Vorce,  C.  M.,  180 

Vosseler,  J.,  604 

Wahnschaffes,  F.,  541 

Wales,  W.,  229 

Wallerant,  F.,  383,  497 

Wallerant's  method   for   measuring   slight 

double  refraction,  383 
Ward,  R.  H.,  226 
Ward's  eye-shade,  226 
Washington,  H.  S.,  292 
Wave  front  of  light,  49,  no 
Wave  length  of  light,  35,  49 
Wave  lengths,  Table  of,  3 13 
Wave  motion,  31 

Wave  motion  in  isotropic  media,  48 
Wave  surface,  Biaxial,  98 
Wave  surface,  Equation  of  biaxial,  99 
Wave  surface,  Equation  of  uniaxial,  75 
Wave  surface  of  light,  49 
Wave  surface,  Uniaxial,  76 
Wave  theory  of  light,  30 


Wedges,  365 

Weights,  Table  of,  623 

Weinschenk,  E.,  230,  231,  360,  456,  538 

Weiss,  C.  S.,  i,  2,  3 

Weiss  parameters,  3 

Wenham,  215 

Wertheim,  G.,  331 

Westphal  balance,  533 

West  rotation  apparatus,  301 

Whewell,  W.,  3 

Wichmann,  A.,  561,  599 

Wiedemann,  G.,  395 

Wiedemann   double    double-quartz    plate, 

395 

Williams,  G.  H.,  590 

Winkelmann,  A.,  442 

Winchell,  331 

Witham,  H.,  572 

Wollaston  lens,  136 

Wood,  R.  W.,  442 

Working  distance,  182 

Wrappers,  608 

Wright,  Sir  A.  E.,  154,  181,  198 

Wright,  F.  E.,  91,  153,  178,  207,  222,  224, 
258,  260,  263,  290,  303,  307,  311,  356,  366, 
367,  385,  394,  397,  398,  454,  468,  469, 
472,  473,  483,  484,  485,  491,  497 

Wright   artificially   twinned  quartz  plate, 

397 

Wright  bi-quartz  wedge  plate,  398 

Wright  combination  wedge,  366,  383 

Wright  double  combination  wedge,  385 

Wright  immersion  fluids,  263 

Wright-Lasaulx  method  for  observing  inter- 
ference figures,  454 

Wright  microscope  lamp,  224 

Wright's  modification  of  Becke's  method 
for  determining  2E,  484,  485 

Wright's  modification  of  Michel-Levy's 
method  for  measuring  2E,  472 

Wulff,  George,  14,  18,  22,  358 

Wulff  net,  17 

Wiilfing,  E.  A.,  26,  27,  99,  154,  237,  284, 
319,  324,  374,  388,  394,  401,  468,  553, 

557,  585,  593 

Wiilfing  projection  model,  27 
Wiilfing  separating  apparatus,  553 
Wiilfing  wall  chart  for  stereographic  pro- 
jection, 27 

Young,  L.  J.,  448 
Young,  Thomas,  30 


INDEX 


649 


Zeiss,  Carl,  132 

Zeiss,  Instruments  made  by,  133,  147,  152, 

184,  186,  187,  194,  208,  209,  289,  298 
Zeloites,  Chemical  reactions  on,  563 
Zirkel'  F.,   252,    260,    284,   557,   572,   574, 

585,  594,  599 
Zone  axis,  4 


Zones,  4,  399 
Zschokke,  W.,  129 
Zwei-und-einaxige  system,  i 
Zwei-und-eingliedrige  system,  2 
(The  last  page  proof  of  this  book  was  read 
December  19,  1913.) 


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